The Rearrangement of Functions and Maximization of a Convolution Integral Citation Welch, Lloyd Richard (1958) The Rearrangement of Functions and Maximization of a Convolution Integral. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/B59P-3R31. https://resolver.caltech.edu/CaltechETD:etd-01192006-081053 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. For the additive group of the real numbers, the value, at the origin, of the convolution of three non-negative integrable functions is less than or equal to the corresponding value obtained from the symmetric rearrangements of those functions. That is [...]. This problem is investigated on general groups and it is shown that under a wide interpretation of the notion of symmetric rearrangement the validity of the inequality implies serious restrictions on the group. Both new and known results in [...] are developed using methods which in modified form yield the answer to the similar problem on the sphere. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Bohnenblust, Henri Frederic Thesis Committee: Unknown, Unknown Defense Date: 1 January 1958 Record Number: CaltechETD:etd-01192006-081053 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-01192006-081053 DOI: 10.7907/B59P-3R31 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 231 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 24 Jan 2006 Last Modified: 17 Oct 2023 18:52 Thesis Files Preview PDF (Welch_lr_1958.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

THE REARRANGEMENT OF FUNCTIONS AND MAXIMIZATION OF A CONVOLUTION INTEGRAL Thesis by Lloyd Richard Welch In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1958 ACKNOWLEDGEMENTS I am indebted to Professor Bohnenblust for the valuable help that he has given me as my advisor. He suggested this problem to me and has given his time and advice very generously during the past three years. I wish to thank the California Institute of Technology for the financial assistance given me during the past four years. Special thanks go to my wife for her patience and encouragement. ABSTRACT For the additive group of the real numbers, the value, at the origin, of the convolution of three non=negative integrable functions is less than or equal to the corresponding value obtained from the symmetric rearrangements of those functions. That is £*g*h(0)<f *g *h (0). This problem is investigated on general groups and it is shown that under a wide interpretation of the notion of symmetric rearrangement the validity of the inequality implies serious restrictions on the group. Both new and known results in E” are developed using metheds which in modi fied form yield the answer to the similar problem on the sphere. IT Tit TABLE OF CONTENTS ACKNOWLEDGMENTS ABSTRACT INTRODUCTION SYMMETRIZATION STEINER SYMMETRIZATION STEINER SYMMETRIZATION ON A SPHERE - APPENDIX 18 a7 INTRODUCTION. Let G be a locally compact topological group whose unit element is denoted by ‘'e'. It is known that such a group has a left invariant measure which is unique up to a multiplicative constant. ! Left invariance means that MA) = M(sa) for every element s of G. The uniqueness of this measure implies the existence of a function A(s) such that as (As) = A(s) Mi). If this function is identically equal to one, the group G is called unimodular. The convolution, f * g, of two integrable functions f, g is defined by £* g(t) =/ f(x) a(x”! t) dx = / f(tx) a(x”) dx If one function is bounded the convolution function is defined every where and is continuous.” We shall be concerned with bounded, integrable, non-negative functions. For the additive group of the real numbers, it is known that for any three such functions f£*g*h(e) < fx g. * h’(e) * where f£ is the symmetric rearrangement of f, and similarly for g and h. 3 The main objective of this study is to investigate this problem on general groups. It is shown that under a wide interpretation of 3) ef. Ref. 3. -2Q =< 'The symmetric rearrangement of a function' the validity of the above inequality implies serious restrictions on the group G. These restrictions are discussed in the first section and a result for the group of rotations in two dimensions is obtained. The second section deals with the group of translations in two dimensions. Although the main result was known, the methods used are more constructive and more direct. Furthermore, properly modified, they are applied in section 3 to a study of functions defined on a sphere. The Appendix contains the proofs of some measure theoretic results which are needed in the main part of the text. First Section. The family of non-negative integrable functions is designatea u'* . Def. Two functions f, g will be said to be a rearrangement of one another if for every 4, M{xl£@) > a} = H{xle(x) > a} ° This notion is obviously an equivalence relation and will be denoted by £f~g On the additive group of real numbers, in each equivalence class of functions there exists one symmetrically decreasing function. This member of a class is called the symmetric rearrangement of any function in the classe On a general group it is not evident whether or not a natural notion of symmetric rearrangement exists. To investigate this question the following two hypotheses are introduced. ~ 3 ww 1+ Hypothesis A. There exists three mappings from L into itself taking £ into f, i = 1,2,3, such that f~fi, f~g implies ft = gt 1+ and for all f, gshelL f*g*h(e)< fl * g* * we) 4 Hypothesis B. Hypothesis A is satisfied by three mappings which are identical. The additive group of real numbers satisfies the stronger hypothesis 3 Be” The additive group of integers is an example where A but not B is satisfied provided the domain of the third mapping is restricted to those functions in pit which assume their maximum an. odd number of times and other values an even number of times.” If a function is in the range of the i-th mapping it will be called symmetric of type i, and if the function is also the characteristic function of a set the corresponding set will also be called symmetric of type ie THEOREM 1. If G satisfies Hypothesis A then G is unimodular. Proof; We first prove that if D A? @,, D C are the characteristic functions of sets of positive measure A, B, C and G is not unimodular then A’ op ,B' B,C! -C 4) For fy, g> h non-negative and measurable, the convolution is defined everywhere provided +o is allowed as a value. Thus f*g*h(e) exists. 5) cf. Ref. 4 and Ref. 5. -i- Secondly we prove that this can not always be achieved by symmetric sets. First Part: Since D1 (x) <1 for all x; 2, * ®., *D,,(e) </ (A) (4(B). On the other hand the upper bound is obtained as follows. By the corollary to Theorem B (Appendix) there exist sets Ay» By with compact closure such that lal (a,) = M(a)s i (B,) = M(B). Define = {al 40a, 27! By) > of . =| awl and the compactness of the closure of this last set Then D C8, , implies L/ (D) < o. By considering a(z") = [(z)]" where n is any positive or negative integer, it is clear that +t can be chosen so that A(t) is arbitrarily close to zero. In particular there exists t such that M (dt) = a(t) iM (D) < M(c) (since Mis) > 0). Again by Theorem B there exists C' such that Dt Cc! and Ad(ct) = M4(C). Now consider @ -1.? a, ’ @., « We have t A 1 Po, “2° “hy, )2@ S, *D ve) 2, “2, re . Using the definition of D this becomes i i a, wD, (x"!y) OD 7) dx dy || Q (x) P,, (x7!y) ax ay K (a) Me) proving the first part of the proof. = 5 = Second Part: Let A, By C be symmetric of type 1, 2, 3 respectively and of positive measure. Hypothesis A implies M (a) (a) =D = D, +, »D. (e) = I. Mal \zB" y dz fous (\27'37!) az) az . On the other hand if 44 (Cc) <U fel AAA N27'p7!) > of then the above equality is obviously impossible. Furthermore to be non=unimodular the group can not be discrete” so that C can be chosen (and thus a symmetric C) so as to have a positive measure which satisfies the above inequality. ( 2| (a (\27'p7!) > Of >O by Theorem Bo.) QED. j4 For the additive group of the real numbers, the symmetric sets are Sy = { xl | x| < 0/2 | and the symmetric rearrangement of a positive integrable function f is defined as follows.’ Define * * a(t) = {xl £() > tt and f (x) = sup { t]x © Sa(t) ‘ » That f is a rearrangement of f can be seen as follows. x eS a(t, ) and t, > t implies f° (x) >t, > +. Thus yS f xi" (x) > tt On the other 1 hand if f" (x) > t then there exists % such that +t, > t= and 1 1 XE Salt y¢ By the monotonicity of the family of sets Sa and the 1 monotonicity of the function a(t), this implies x ¢€ Sat)? Thus Pa(t,) c { xl£"(x) > +} c Sa(t) and a(t, ) < ufzlt (x) > t < a(t). 6) Haar measure on a discrete group assigns unit mass to each point and thus is right invariant also. 7) cf. Ref. 5s Pe 276.6 -~6- Since a is monotone decreasing and continuous on the right, a(t) =f x|f (x) > t | which proves that r is a.rearrangement of f. That Hypothesis B is satisfied for this symmetrization is a simple consequence of the validity of the inequality for characteristic functions of seta.® The above discussion need not be restricted to the real line. If there are three monotone classes of sets such that Hypothesis A is valid when pit is replaced by the class of characteristic functions of sets then it can be extended to y'* by the above construction. Further- more it is a consequence of the corollary to Theorem C (Appendix) that the symmetric functions given by Hypothesis A differ on sets of measure zero from those constructed from symmetric sets. The next theorem shows that if Hypothesis A is valid on a group the three monotone classes exist. THEOREM 2. If Hypothesis A holds the symmetric sets of type i form a monotone family of open sets to within measure zero. (That is, there exists a family { Orla & range of Ht such that a < 8 implies or Con ’ KE (o,) =a and Mis. @ o7] = 0. Proof: The unimodularity implies that cyclic permutations of the three functions in the double convolution leave the value at the origin invariant so that any cyclic permutation of the three symmetrizations also satisfies Hypothesis A and it will suffice to prove that the first symmetrization has the stated property. 8) Cf. abide, pe 279-280. ~Jo rst suppose G discrete, then we can normalize the measure so that each point has one unit weight. Let C be symmetric of type 3 and Measure one and let B be symmetric of type 2 and measure n. Then ) -| -{_-1 * = = e qd. «D, De) on Atat\ya') ay = Ufa Me7'37"] It is obvious that if A is symmetric of type 1 and KM (A) =n then a=c'g7! in order for the maximum to be achieved. Likewise if 1 ana Ma) <n then ACec 'B™! ana if fA) > 2 then aDew'p the monotonicity is established. Notice that the proof in the discrete case made use of sets of the form { 1D, * D, (x) > +} to separate A's with different measure.” This same method will be used in the non-discrete case. Let Ais A) be symmetric of type 1 with (/ (ay) < May)» Since G is not discrete there exists 45 such that f4 (ay) < LU (Ay) < lag) 10 Furthermore by Theorem B (Appendix) there exists Aas Ag such that A, = A, Ca, CAs Af(Ag) = (Mas) and {4 (A3) =U (Ay) Now let g be an approximate identity for ® - that is, g > 0, / g=1 and 3 le +. “Pall < Finally let F(x) = Pa «D - (x) and ty. tos ty be the numbers A 3 whose existence is asserted by Theorem C (Appendix). Then, to within 1 1 1 11 measure zero A, Fs Ay DF, » Ag OF, ,and t,>t,24,- 1 2 3 9) To separate in the sense that if A <B<C the B separates A and C. 10) Non-discreteness implies there are sets of arbitrarily small positive measure (see Ref. 6, pp. 123-124) and the assertion then follows from Theorem B. 11) See Theorem C for definition of Fy - 8@ Dt Also by Theorem C if t, < ty then F i to within measure zero t 2 and AD ~ Ay which proves monotonicity to within measure zero. That € can be chosen so as to make to < t, can be seen as follows. By unimodularity, Q, x g «OD _, (e) -D D0) = | (g “O. - O- *Q, ©) . Ag Since (g ‘PD ., -D.) * ® (x) is continuous [see footnote 2] 3 3 i supremum of absolute value and || ||, are the same so that lI sn) GQ) 1D *e*®_e)-O *O_ els tle*O_,-O)* , i A, i Ag Ag Ag 1 The last inequality is the theorem in Ref. 6, page 121. Now ®, * ? (e) = Uta, May) = min[{d(A,), 44 (A3)] so that (1) implies min[ (A(A,)s LA (A3)] - © <P, * 8 *D _(¢). 3 Symmetrization does not decrease the double convolution so that (2) min[ (Ay )+ (43 )] -e< @.; * Pes *D?_, (e) < min[ L(A, )sf4(A)1 | 4 3 The last inequality is obtained from the facts that 1G = 1, and © A.’ D -; < 1. Now by Theorem C (Appendix), i A 3 ; -~9-= (3) DO 1” eg" D>, (e) = f F(x) dx -f F(x) dx + [MG, )- MF, Nt; . Ay Ay val Fe i i i t, = +t. then by (2) and (3) applied to i =1 and 3, If t 2 3 , = A.) = =@— * g* xT) _(e) - * gm * [U(ag) ~M(a, It, il g Px ®,; g Pa > (Az) -[A,)-e while (2) and (3) applied to i = 2 and 3 yields [Mla,) Mgt, =D , * &F *P (0) - ©, rE *D edce, 3 Ay 3 3 If ¢ = (1/2) minf[ (a3) - HMa,)s [Al4g) ~ Mia )] then the inequalities < ty. ean not be simultaneously satisfied, so that it must be that te { F(x) is continuous [see footnote 2] so that Fy, 2 1 1 ; os we have shown that if bé (A) < Ma ) then there exists an open set, 0, is open. Thus which separates Ay and al to within measure zero. Choosing a sequence Al such that Ma)4 i (a!) shows that U 0,C a’ to within measure zero and MU 0, ) > Ma.) 4 (a!) Thus 0, is i i an open set which differs from al by a set of measure Zero, QeBeDe 1+ Theorem 2 says that any three mappings of L into itself for which Hypothesis A is valid is not essentially different from symnetrizations based on three monotone classes of open sets. The proof can be made to yield more in the discrete case, namely, Hypothesis A 1-1 where implies Hypothesis B. For the proof shows that Ay =c By -10- where {|} = Cy. Applying this result to the cyclic permutations of the three classes gives Bo = av'on! 2 C= pla! from which it can be shown that Ay a! = aBic = ov! Cue Furthermore unimodularity gives QD -i * ® * QO-,© =, * ®, «D,e) . So that the symmetrization based on the class ana”! shows Hypothesis B to be satisfied. It is conjectured that in general Hypothesis A is valid if and only if Hypothesis B is valid. A further restriction on any G for which Hypothesis A is valid is the following. THEOREM 3. If there exists M and Ny, subgroups of positive measure, such that 4 (M) < (N) and Mw WAC) is not an integer then Hypothesis A is not valid. Proof: We assume Hypothesis A and begin by showing that ul and w! are “almost" subgroups. Now @,, * «OD, * + Dee) = / / 0,097") dx dy = [ 4 (u)]? andy . M wl i since this is also an upper bound over all rearrangements, -1 / Mhenxe) J] ax = oD, * ® 2 * ® 5(¢) = [eny)* M Since HIM x0 "J <A (M), then ulc CM= { leat?) =n} except for a set of measure zero. Otherwise the upper bound could not be attained. On the other hand -11@ Mm) = ivaal (MP) dx </ {A OPM x00? )71] ax = [cacy]? 5 so that MM (M) = MM and ul =M except for a set of measure zero. 1 Next it is claimed that if me M then Mm isa group, which will be shown as follows. Now the measure of the symmetric difference, WO x(w)"', is LU {?@x02 7} = 08)" } + Afoey} - 2pipenntw? yt, But unimodularity implies LL {or yi { =A { we) } = { oe f = kwh ° Thus x eM if and only if MM w@ x()"} = 0. It follows from the triangular inequality for the measure of the symmetric difference of sets that x ¢ Mn”! if and only if {xr @ nt” f= 0. Finally, if xs ye Nin”! then i (2 @ xy! Mw) < (wr @ xi’) + (x? ® xy! we) =H (wf @ wi) + (yu @ v*) = Oy 1 1 so that xy ce Mm. Which proves that Fin”! is a group. Similar arguments hold for H. Since M’ Cn! and M,N differ from om’, ni by sets of measure zero, we have <U(MM\N) = /A(M) > 0 and so MON is non-empty. Let me M Ny then Tin and. fin! as alsois (M{\N) m™!, Further, (4 {@aR) n7} = (4(M). Now fin”! consists of a union of cosets of (MN) m7! go that Min) = in) are subgroups of G = =| = fA (M) times the number of cosets of (MN) nv! in Nm. But this contradicts the condition of the theorem that / (N fA (M) be finite but not an integer. QeleDe -12- Theorem 3 eliminates from consideration any finite groups other than those whose order is a power of a prime. The latter case is Still open. Examples. The group of translations in Euclidean n=space is an example of a group in which Hypothesis B holdse The case of the real line has already been cited. The higher dimenionsal cases can be obtained from an extension of the work of cross !* (or from later parts of this paper). Gross constructed a suitable Steiner sequence !? in 3=space such that beginning with a given integrable set the successive sets converge to the interior of a sphere. By convergence we mean { (a1 @ S)>0 as i >o, where © indicates the symmetric difference of two sets. This together with Theorem 5 shows that Hypothesis B holds where symmetric sets are the interiors of concentric spheres. It is conjectured that Hypothesis B holds on the circle. However the following theorem is the best result available. THEOREM 4. Let f, gs h be integrable functions on the group of plane rotations (represented as { al-1 <@< nv} ). Furthermore let QO = {a ig| < a/2} for O<a< 2m and e! be the symmetrization a of f based on the class { 0, ; and similarly for a’, n!, 14 Then e*g*ni(o) stl «g! «n'(o) 12) Ref. 7. 13) See Definition preceding Theorem 5. 14) pl is also called the symmetrically decreasing rearrangement in analogy with the corresponding rearrangement on the real line. -13- Proof: By the remarks preceding Theorem 2, the general result follows from the validity of the inequality in the case where f, g, h are the characteristic functions of sets. Let A», By C be integrable sets, with measure 2a, 2b, 2c, respectively (for convenience pb (G) = 2m) and define 2() =, «Diy Then ®, * P, * D (0) = [ F() ag . Theorem C (Appendix) asserts the existence of t such that i O,) <2as Ul dir) 2 tt where B, = {ale(é) > sf . For each a designate by t(a) the infimum of all such t. Notice that F'(g) = t(|g]) and [v0 ag < [ A F “a | F(B) ag + t(a)[2a ~ LF, (, ))] -[ P(g) ag. t(a) ~ a, The proof of the theorem reduces to showing a a 1 [ F!(d) ag </[ OD! «Di (-g) ag ~a “a at this point we introduce Lemma 2 which will be proved later. LEMMA 2. |t(a,) - t(a,)| < la, - a| -14- Let us now examine FL) = pap, D4 * DU(-0) af ~ min[2b,2c] for |Q| 1A |b = c| (1) F (¢) = 4+ max[0, 2b + 2c - 20] for {dl iv min[bte, 2a-b=c] -linear and continuous for all g * . In order for F to be linear and continuous the derivative must be -sgn f for |b=-c| < || < min[btc, 2-b-c], With the use of the definition of F and F' 4t can be show that (2) min[2b,2c] > F! (g) > max[0, 2b+2c=27] and ns 1 7 * (3) / F' (g) ag = f[ F (f) ag = dbe. = =i If a< |b-c| or a> min[btc, 2n-b-c] then the inequality [i r!(g) ws [re ag readily follows from (1), (2) and (3). Now suppose ri(a) < F(a) and | lb-c| < a < min[bte, 2i-b=c] . Then by Lemma 2 F'(\x|) - Fi(a) < Ja - | | so that for |b-c| < |x| <a - 15 = Fi(|x|) = Fi(a) + Fi(|x|) - F(a) < F(a) +a - |x| = h(x). Also F! (x) < min[2b,2c] = F (x) for |x| < |b-c| a % | [ [F (x) - F (x)] dx>0. “a On the other hand suppose so that *% F (a) > F (a). Then again using Lemma 2 for a < | x] < min[bte, 2u=b-c] gives #! (x) Fi(a) + F(x) - F(a) > FX(a) = (|x| - a) = F(x) Also F! (x) iv max[O, 2b+2c<2n] = F’ (x) for min[bte, 2mbec] < [- [Le (x) = Fi (x)] ax = _[ (F(x) - Fl(x)] ax > x A cole so that Vv Oo QebeDe In what follows D., D.. F, F will be assumed to be periodic functions on the real line with period 2m. To prove Lemma 2 we shall need LEMMA 1. | F(x,) - F(x, )| < |x, - Xo . -16< Proof: We can obviously assume x, 2 X5¢ Then — 2 | F(x, )- F(x, )| = [Po Oy, -y) ay - [O01 -y) ay wy “X5t -|[" o, (7) ay - a Du) a -|[* (Dv -)- By +e) | =, x 2 1OxG -c) -D. + c)| dy “xy 1A 1A X47 XQ. LEMMA 2. |t(a,) - t(a,)| < la; - gl - Proof: We may suppose t(a, )< t(a, ) and thus a, > aoe The proof consists of constructing two intervals in Fe (a OVE, t(a 3)? where A a4 denotes the complement of A, and using Lemma 1 to estimate their length. Suppose O<a, and a, <a. Then by Theorem 6 (Appendix ) Hf falP) > tay) } 2 2a, > 0 i = 152 and ff xlF(x) < e(a,)f Since these sets have positive measure they are non-empty. Furthermore an “MH Fe(a,)? 2 2m = 2a, > O- Lemma 1 shows F(x) to be continuous. Thus there exists X59 i = 1,2 such that F(x,) = t(a,) . -17- The cases a, = O or a, = 7 can be handled by taking limits and pointing out that the circle is compact. Since translations of F do not affect the statement of the lemma we can assume X4 =f and =-T < Xo < We Let B, = sup {xl-" $x< x, and F(x) = t(a, y B, = inf { «Ix, <x<tm and F(x) = t(a, } . Both of the sets exhibited are non-empty since F(-0) = F(a) = t(a,) so that Bb,» B are finite and F(6,) = F(Z.) = t(a,) by continuity of F. Next let Bs inf { ald, <x$x, and F(x) = t(a,)} i Bs = sup { =lx, <x< B, and F(x) = t(a,)} . A similar argument shows that Bis Bs are finite and F(64) = F(3) = t(a5)6 Now x & (B,» £3) U(d5, By) implies t(a,) < F(x) < t(a)) which tells us that 4 x| F(x) > t(ay)} » ty» B1)s (Bt, dp) are disjoint and their union is contained in F, (a,)° Then it follows from the 1 definition of (ay) and t(a,) that By Lemma 1 so that (1) becomes ). QekeDe alt(a,) - t(a,)] < 2(a, - a, -18<- Second Section. Steiner Symmetrization. By the direct product of two groups, G = Gy x Go» we mean the set of all ordered pairs <6, ’ 85? with gs & Gs and multiplication defined by <8y: Bo” <hy » h,> = <g4hys Boho? « The groups Gy» Gp are isomorphic to subgroups of G, namely G, ~ G, XK @y Gp ~e x Goe Here~ after G, 2 Go will be identified with these subgroups and two decompos= itions will be considered distinct if at least one pair of corresponding subgroups are. If Gy» Go have left invariant measures, M 1 and & 2? then it is obvious that the product measure is a left invariant measure on a. !? Now let G=HxR where R is a subgroup identified with the real numbers, and let /, ’ as 2 be the left invaraint measure on H * and Lebesque measure respectively. MM 2 denotes Lebesque outer measure. DEF. If A isa measurable subset of G then generalized Steiner symmetrization of A with respect to the subgroup H is , * Ay = {cn ye>lh eH,reR and Ir] < $M fel<ayr> € a3} * rf ei 2 |r |<hye> & af =O then the condition Ir| < 3 Ml, {rl<asr> € af is replaced by the condition r is an element of the empty set or r=0O accordingly as { r|<h,r> € a} is empty or non-empty. In more descriptive terms, generalized Steiner symmetrization replaces the intersection of A with the line hx R by an interval with the same linear measure and centered at <h,0> . By Fubini's 15) For definition of product measure see Ref. 6, ppe 44-456 = 19 - theorem { (A,,) = f(a) so that A. is a rearrangement of A. H The group of translations in Euclidean n-space is the n-dimen- sional vector space over the real numbers and admits of infinitely many representations of the form H, x R, where R, is a linear 1-dimensional vector space (and thus isomorphic to the real numbers) and Hy is an n=|-dimensional subgroup. If we introduce a scalar product then we have available the concept of perpendicularity so that the following definition makes sense. DEF. If Vn is the group of translations in Euclidean n-space, H is an (n-l)-dimensional subgroup and R is a 1-dimensional subgroup perpendicular to H, then the generalized Steiner symmetrization is ealled the Steiner symuetrization. '© Since generalized Steiner symmetrization takes monotone classes of sets into monotone classes of sets, the Steiner symmetrization of non= negative functions can be defined by means of the construction of symmetric functions in terms of symmetric sets which follows Theorem 1. DEF. If {x,§ be a sequence of (n-l)-dimensional subgroups and A an integrable set in Vn and if a° = Ay Atos Gi), » then fat} a is a Steiner sequence. THEOREM 5 Generalized Steiner symmetrization never increases f* g * h(e) provided f,; g, h are non-negative integrable functions. = 20- Proof: Let G= G, xR. s and t will be elements of G, and x and y will be real numbers. f * g * h(e) -[ J al f£(s5x) a(s 'tey-x) a(t! yy) dx dy ds dt, Gy G, RYR The order of integration has been rearranged using Fubini's theorem. Now for almost all s and ty f(s, ); a(s 't, ) and a(t”, ) are measurable functions on R so that the validity of Hypothesis B on the additive group of real numbers implies for almost all s and t If f£(s5x) a(s” 't,y=x) h(t s~y) dx dy < RY R < [J £!(syx) gl (s7't,y-x) hl (7! ,-y) ax ay. RY R el(s, ) is the symmetrically decreasing rearrangement of f(s, ) and similarly for a(s 't, ) and n(t!, ). Integrating both sides of the- above inequality over G, x G, and observing that t! (s,x) is in fact the generalized Steiner symmetrization with respect to Gy gives the theorem. Qe HeDe Steiner symmetrization was originally defined for convex sets in the plane in connection with the isoperimetric problem. It was shown in the reference in footnote 16 that Steiner symmetrization does not increase the perimeter of convex polygons. More generally it can be shown for any polygonal Jordan curve, I, with a finite number of sides that the symmetrization of the curve and its interior results in a point set whose boundary is a rectifiable curve whose length is not greater than the length of [T. (It will not necessarily be a Jordan curve.) ~21- By suitable polygonal approximations the results extend to more general rectifiable closed curves. However, the following theorem has a direct proof, THEOREM 6. Let I be a rectifiable closed curve in the plane such that the winding number of every point not on the curve is 0 or 1/7 Let A be the set of all points which lie on [ or whose winding number is 1. If Ay is the Steiner symmetrization with respect to the line L, then the boundary of AL is contained in a curve whose length is less than or equal to the length of [’ and which has the same winding number property as T. The proof is very similar to the proof of Theorem 11 and so will be omitted. The rest of this section will be restricted to Steiner symmetrization in the plane although some extension to higher dimensions is possible. Let jus} be an arbitrary sequence of one dimensional sub= groups of Vo and {7} the perpendicular subgroups. Further let A be a set with a rectifiable, connected curve for its boundary and {7} be the Steiner Sequence generated by A and {15} « if we rotate the sets at such that their axes of symmetry are all Ly» then each at ean be represented in the form At = {<x,y>|<x 50> & Ly <Osy> © Ry and ly| s F, («)} . 17) The winding number is also called the topological index or merely index. For the definition see Ref. 9, pe 149. ~ 22 = For the perimeter to be finite and connected, A must be bounded, ic@e, there exists M such that <x,y> ¢ A implies x” +y* < we Therefore a F, (#4) = 0. The perimeter of a* 4s abound for the variation of F; and this is bounded by the perimeter of A by theorem 6. Thus the functions By By Helly's selection principle there exists a subsequence which converges have a uniform bound on their variation and F, (4M) = Oe almost everywhere to a function F and the corresponding subsequence of sets converges in measure to the corresponding set. It is not difficult to show that even without rotating the at to a common line of symmetry there exists a subsequence which converges in measure. The following theorem shows that the restriction that A have a finite perimeter can be weakened. THEOREM 7. Let Lys Rys Lys Ros Fy» Fy be as before and also L, 7 Lye If A is a bounded integrable set bounded by M then Fy is of bounded | 2 variation and the perimeter of A™ is finite. Proof: We choose two coordinate systems and an orientation in the plane so that L, = {tax dlyt = of ’ L, = { Govly = of and the angles; ad, from Ly to L, is positive but less than or equal to 1/2. a First suppose a= n/2, then the second symmetrization is seen to be the same as replacing al by the area bounded by the symmetric decreasing rearrangement of Fy (x'). The symmetric decreasing rearrange ment is monotone increasing in the second quadrant, monotone decreasing = 23 - in the first quadrant, F (-M) =F * (Ma) =O and F (O) < M. Thus var F < 2M. We can define the perimeter as n 2 sup ») "V (x. <x i=] 2 * * a) gyn + (FG) - Fy). The supremum is taken over all partitions of the form -M = X<X, eoecy nile A bound for the perimeter is 2 var F + 4AM < 8M. Next suppose 0<a< 7/2. Let KM; be Lebesque measure on the line and define Foy (x) =A, { yl (sy) eA, and y > x tan a} Foo (x) =M , { yl (xsy) © A, and y < x tan a} . The above exhibited sets can be shown measureable for all x and it is obvious that F(x) = Foy (x) + Foo (x) ; The function Foy (x) + x/sin a cos a will now be shown to be monotone increasing. Suppose x, > X59 then X47 Xo Poq (x,) ~ Pot (x5) + sin a cos a oo “eX D Aha sy) dy - Jean «®) (x5 sy) ay + at 2 0. Xy tan a | i a (D, wis Di 9 7 "3 ta ad ad tan a +O, “x p/sin a cos . a) ~ x.tan & (x59) dy + sin a cos a (1) 2 > f - (Dj) -D Oper + 2D) ay , 1 x,tan a ~ 24 = xy “X5 i Ch) 1 2 p Now if Q, (x, > y +e -t atn2) = then the point Ff (x,9 ytora Z.) is in al and its primed coordinates must satisfy (2) ly"| s FG) where X4 7%, x! =X, cos a + sin aly + p= a) = cos a+ y sina X47Xp X47Xp y' = -x, sin a + cos aly + p= a) my sinaty cos ators Expressing (2) in the unprimed system gives x “Ky Sina 2 yy cos a+y sina). (3) -x, sina +y cos a + The last term on the left hand side of the inequality is positive and, in the range of integration of (1); “x, sin a+y cos a is non-negative so that in the range of integration (3) implies |-x, sin a+y cos al SF, (x, cos a+y sina). But this says that oF 4 (%, vy) = 1. We have just shown that in the range of integration Q +i» ytx 47Xp/tan a) = 1 implies oo) 1 &, sy) = 16 Therefore the integral in (1) is non-negative and the aomotent od ty of Fo4 (x) + x/sin a cos a is established. Furthermore Fy om) =F, 4 = 0 so that F,, (x) +x/sin a cos a is of bounded variation in a. A | =F similar argument is valid for F, so that F 2 + Foo is of bounded 22 2 variation. Defining the perimeter as in the case a = 1/2 completes the theorem, - 25 - The existence of subsequences which converge naturally raises the question, "Under what conditions does a Steiner sequence converge ?* That not every Steiner sequence of a bounded integrable set converges can be seen by the following example. lIet the angle as between Lo and L 1 be i/n and let A be a circle whose center is on Ly but not nt <0,0> . Then if ry is the distance of the center of A from <0O,0> n= n ., : - then A is a circle whose center is at a distance rr =r, i cos 4, from <0,0> . The infinite product converges absolutely to a non-zero ew limit and the lines L tn intersect the circle Vx" 4y* =P, cos a, i=i in an everywhere dense sete Therefore the sequence of centers have all points of the circle as limit points and at does not converge. iss) ow A * : i " 2 -— ° bond It is conjectured that if 2, Ja. <@ or py a) = & and if A is bounded then the Steiner sequence converges in measure. The following theorem will enable us to show convergence of Steiner sequences provided the sequence {x44 is periodic. THEOREM 8 Let A be a bounded integrable set in Vos 12, a sequence of one dimensional subspaces and Rs the perpendicular subspaces. Then the Steiner sequence obtained from A and {1} has the property that . < +13 2, Ma ee yy <a , n=1 The proof will not be presented here since it is analogous to and some~ what simpler than the proof to Theorem 12. COROLLARY. £4 (a @,™"N) 59 as n>. The proof is obvious. - 26 - THEOREM 9. If the sequence {1} is periodic and A is bounded the Steiner sequence converges in measure to a set having all the Lo as lires of Steiner symmetry. Proof: Let {ts} have period p. Acearding to Theorem 7 and the discussion preceding it there exists a subsequence A k converging to a set A’ At least one of the subgroups Las ooo 9 Los say Ls must occur infinitely often among {t ' and it can be seen that a” must be Steiner symmetric with respect to L, to within measure zero, Now consider the sequences Ma, tN must occur infinitely often in this sequence. But since Mar @a for N= 15 ces , prt. The line Loan nt) > 0 as no, this sequence also converges to A’ Thus a’ is Steiner - symmetric to within measure zero with respect to all the lines {us} e It can be shown that MA @B) = MA, OB)» that is, Steiner symmetrization does not increase symmetric differences. This fact coupled with the fact that A. is invariant under symmetriza~ tion with respect to all {1} proves that the sequence at converges. QeBeDe Polya and Szego point out in Ref. 11 that a Steiner sequence in 3-space based upon two planeswhose included angle is an irrational multiple of wm must converge to a solid of revolution about the line of intersection of the two planes if the sequence converges at a1, '® 18) They also point out that the "if" has been removed in the case of convex solids. Cf. Ref. 10, pp. 86-90. -27 - By observing what happens in each plane perpendicular to that line of intersection and using the preceding results, we can remove the "if" for all bounded integrable sets in 3=space. Third Section. Steiner Symmetrization on a Sphere. Although the sphere is not a group manifold there is a symmetrization analogous to Steiner symmetrization in the plane and several of the theorems carry over to the sphere. In particular, the isoperimetric property (Theorem 6), the convergence to zero of symmetric differences in a Steiner sequence (Theorem 8) and the convergence of a Steiner sequence based on a periodic sequence of coordinate systems (Theorem 9) all hold on the sphere. On the other hand, Theorem 5 (the decrease in the value of the convolution of three positive functions ) must be modified before it is valid on the sphere. Let GC be a spherical coordinate system. That is, C consists of a mapping P(@,@), which maps [- 5 ’ 5] X [=-7, 1) onto the unit sphere such that the element of area, dP, is cos 8 de dag. (@ is the complement of the polar angle and g is the azimuthal angle.) We will designate Lebesgue measure on [-n, a) by i 1 and Lebesgue outer * measure by Mi ° DEF. Let A be an integrable set on the sphere. Define - 1yy* 7 bog = {Al Isl s 3M] {[olrtes) & 3} if { dlp (0.8) E at #0. Otherwise define Ao.e to be empty or {0} accordingly as {d|P(6Pe A } is empty or not. - 28 = DEF. A, = { P(e.8)|d E Ag gf : In attempting to construct a theorem analogous to Theorem 5, two difficulties occur. The sphere is not a group manifold and the necessary property of the real line is replaced by Theorem 4 which applies to the circle. Theorem 4 is adequate for the following theorem and the first difficulty is avoided. THEOREM 10. Let e (P,Q) be the spherical distance from P to 4, and let H(P ) be a monotone decreasing function of CP . Further let F, G be non-negative integrable functions on the sphere. Then the Steiner symmetrization of F, G does not decrease I= J [xe G(Q) H(Q (P»Q)) dP dQ. Proof: As before the general case follows from the particular case where F and G are the characteristic functions of measurable sets on the sphere, and H is the characteristic function of an interval 0 <0 < he Writing the integral in the coordinate system C and interchanging the order of integration u/2 0/2 Tw wT of tf f. F(@,%) G(@!,f') H( QP (0,0! wo")) ag ag] ~a/2 top ea Jen *cos 6 cos 6' de de!, Consider H(O(050,0',9')). For fixed @, @ this is the characteristic function of the spherical dise consisting of all points (8',f') whose distance from (@,4) is less than or equal to h. The circle 6' = constant either does not intersect this region or intersects in a connected arc of the form [fg - a, $+ a] where a is a function of 6, 6'. Define Hy jot (2) to be the characteristic function of the set, {gl |\g| < a}. Then 12 [e cf " F(@,0) G(e',¢') # (p-2') ag ag) -a/2 J -0/2 oat J -1 98 *cos 6 cos 6! de det, as = vane 8. ww(\t 1 sy Since Hy gr ) Hg ot ( x) and G(6', -g*) isa rearrangement of G(6',f'), Theorem 4 applies to the bracketed expression and yields u/2 1/2 tt my 1 / I / F' (98,8) G (9',6') H(P (P3Q)) 0/2 J-a/2 J -1 J ma «cos @ cos ef dg ag! de de! , where 1 af {gl «< 5M, { al F(@,8) = if #! (6g) = 0 otherwise and a similar expression for al Obviously yl is the Steiner symmetrization of F with respect to C to within measure zero, and the theorem is proved. The statement of the analog to Theorem 6 is somewhat complicated by the properties of the sphere. Let the sphere be represented in Buclidean 3-space by z = sin @, 2 = cos 8 cos J, y = cos @ sin g and consider the stereographic projection of the sphere from (-1,0 0) onto - 30 = the yz plane. The winding number of a point of the sphere relative to a curve bounded away from (-1,0,0) will be defined as the winding number of the stereographic image relative to the image of the curve. THEOREM 11. Let C be as before and let [. be a rectifiable curve, bounded away from the point whose coordinates are (O,-1), such that the winding number of every point not in [ is either 0 or 1. Define A to be the union of [. with the set of all points whose winding numbers are 1. Then the boundary of Ag is contained in a curve [' whose length is not greater than the length of [ and which has the same winding number property. Proof: I is a closed set and the limit points of a set of points with 19 Thus winding number 1 either have winding number 1 or lie on I. A is closed and { B|P(@,f) « at is a closed set on the unit circle and so is measureable. Thus #(@)>BF $M, { g\P(e,g) € a exists for all ©. Define @, = inf {9| Ap such that P(0,f) eT F 8, = sup of the same set. Now if the cap, { P(6,A)le < 8, } » possessed both points in A and not in A, there would be points of [ with @-coordinates less than 84° Thus the caps, { P(8,f)|@ < e, / and { P(o,d)|e > a, } consist entirely of interior points or exterior points of A. By the definition of Steiner symmetrization on the sphere this implies each cap is either interior or exterior to Ag so that are has no boundary points with 19) Cf. Ref. 9. ~ 31 - A simple argument shows that the boundary points of Ag are contained in B where B= { P(e.) le, <6 < 65, min[F(6), lim F(x)] < [dl x8 < max[F(@), lim F(x) f , xe In fact the only additional points this set has are of the form (6,7) and are thrown in to make the boundary connected. — oa Define L(@) = be » V (x,y )* + cos* ¢ jl F(x, )-F,_, 1" where lim is taken over all partitions, 0 = fx, < E <x, = 6, < <& 2 < Xo coe < x, = @ and lo| = me (x, - Xi) In the event 8, = -1/2 the first term is omitted. It is obvious that we are trying to "measure" the perimeter and the first term nicely takes care of boundary points of the cap, 6 < 8) whether it consists of interior points or exterior points. The cap 9 > 85 L(@,+) = lim L(x). The + sign is omitted if ©, = a/2. First we show x7o+ that 2L(@,+) is less than or equal to the length of T. Lastly we will is taken care of by using exhibit the curve I which has length 2L(8,+)» maps onto By, and has the right winding number property. Proof that 2L(8,+) is less than or equal to the length of IT, The following lemma is used and its proof will be found in the Appendix. - 32 - MMA TO THEOREM 11. Let TI, F(@), O be as before and let [ have length S and arc length s as parameter. Let 1,(0) = 4 slay, < @(s) < x, oF e(s) = : and ag > Oo = ag or @(s) = x; and as * ) Then @(s), f(s) are absolutely continuous functions of s and 2|F(x,) - Fay )1 s i © ie ds. i The lemma is used as follows (1) 4 cos” & lF(x,) - F(x, _)1* < cos® E ho ad) as}? < fe cos @(s) Es ds)” I, (0) + 2(x,-x wth soos e(s) sf 5 ds] ag | if oll Fron (1) it follows that (2) 2V[x,-x _1)*+L F(x, )-F( -| 1? eos” € < Valx,-x,_,] Uf cos 0(s) Ea as]* + (0) os “x, 4) as a @(s) ise | wt a 8) 35) Val, 4) 41 cos 0(s) [$I = I, (9) oo» 33 « The last term is less than Io} (64a - If a>O and we I; restrict the partitions to these for which =u /2 +a < ae and x < n/2—a, then summing (2) over 1 < i<n gives (3) PD» 2 Vix, “x, _)° + (F(x, JoF(x, J? cos a < x 2 Vall fe cos e(s)|$ af as}* + lol. s, If Q, > -a/2 and @< 1/2 then (3) implies d 2 (4) 2L(@) s ee z Vala, ) 741 / (or e(s)[$4| as]* . The cases e, = -1/2 or @= 1/2 can be handled by taking limits. Now if 8, <x <x, S 8, then the fact that [ is a simple connected i-1 2 closed curve and the definition of 8, and 8, two disjoint subsets of [0,S),) say 1, (0)s Ti (Os such that X54 <x implies x = @(s) has a solution in each set. Since @(s) is of imply that there exists bounded variation and uniformly continuous, Ix; ~ x; - < Var e(s) = ly (0) -[ Fe ds with a similar expression for (a). Also 11 (0) (9) | and L(o ) are contained in 1,(0) so that ae 2|x,-x . | [| |=| ds Pee ST] oO Finally (4) becomes (5) 2L(8,+ §) n Tin 7 98) 4 “ @ ia 2 s28 +t zy vl (a) ]*+[ bo (s)[$Z|as}* | The 2 § is the contribution to |x,~x,_;| from that portion of the partition between 8, and Se, + § » By Minkowski's inequality~° n (6) 2L(0,+§)< 26+ lim f yi se + cos” @|22\? ds are lal-o &, (0) 9 ds Since the 1,(0) are disjoint 2L(6,+ §)<28+S. Taking the limit as $ -~ 0+ completes the first part of the proof in the event 6, < a/2. If 6, = 1/2 then it can be shown that 2L(8,) = 2 2 2 lim L(x) so that a slight modification of (5) and (6) corresponding x78," 2 to § <0 gives the result. Lastly, B may be represented by the points of a curve of length 2L(@5+) as follows. L(@) is a strictly monotone increasing function of @ so that for O<s < L(@,+) there exists a unique 6 such that L(@-) < s < L(@+). 20 Ref. 5, formula 6.13.2. = 35 = Define 8'(s) tobe that 6 If s< i(@) where @=6'(s) define A¥(s) = - [ r(@-) + PLE AE} (5 - 1(6-)) and if s>L(@) where 6 = 6'(s) define ér(s) =~ [ r(e) + SCH = FS} (g ~ 4(@)) 1. This represents a curve I°' in the negative hemisphere (¢ <0). Re= placing @'(s) by 9 (2L.(@,+) ~s) and f(s) by “B (2L,(8,+) - s) completes the curve, ['', in the positive hemisphere (f > 0). Notice that @(L(@,+)) = 6(2L(@,+) = L(®,+)) = 6, and (L(8,+)) = F(6,+) while ~§(2L(8,+) - L(@,+)) = “F(@,+). Now the cap 68> 6, consists 2 either entirely of interior points of A, or of exterior points so that C F(®,+) =O or a. In either event [@, sF(@,+)] and (@, -F(@,+)) represent the same point. Thus the two pieces of [ are connected so that I‘! is a curve. A similar argument at 9, proves that [' is a closed curve. We must show that I‘! has the winding number property. That is exterior points of A. have winding number O and interior points C have winding number 1, If both caps, (6 < 8,) and (@> 85) consist only of exterior points then the mapping I! +T! gives by 6'(s)96' (s) and §(s) > (1-0) f(s) is an admissible homotopy relative to exterior points.*! For a=1,T! lies entirely on the arc, J = 0 and the winding number of any point not on ry is Q. If a cap is interior then construction of the homotpy must be preceeded by a homotopy 21) See ref. 9 for definition of admissible homotopy. - 36 - which frees the cap from the curve. For interior points the argument is as follows. (6,96,) is interior to A, if and only if 1d, < min{lim F(x), F(6,)]. But if C x79 (8, »8,) has this property then so do all points of the arc |d| < Ab 15 6 = 85° Since winding number is continuous ~ this implies that (0,99) has the same winding number as (8, 28,)« Now examine the stereographic image of I[‘' in the yz plane. The curve goes from the image of (6, »sF(@,-)) to (0, 2F(@,+)) in the negative half plane (z < 0) and back again in the positive half plane. It is easily seen that the net change of argly + iz - Yol for Vo between these two points and not on the image of [' must be 2m so that the winding number is +1.19 The analyogy between Steiner symmetrizations on the plane and sphere breaks down at Theorem 7 where a finite perimeter was obtained. The proof of Theorem 7 depended upon the fact that lines perpendicular to I, meet the lines perpendicular to L, in a positive angle. How= 2 ever if C C' are two coordinate systems on a sphere with common origin (P(0,0) = P'(0,;0)) then the lines of constant 9 are tangent ‘to the lines of constant @' if they meet along the great circle g= + u/2. It seems evident that a set could be constructed which was Steiner symmetric in C but which had infinite perimeter in every neighborhood of the great circle ~=+ ,/2 and such that A, also had infinite perimeter. This is apparently the only point where the analogy breaks down since we haves THEOREM 12. Iet { c5} be a sequence of coordinate systems on the sphere with common origin (P, (0,0) = P,(0,0)) and let A be a measure- - 37 = able set and {3} the Steiner sequence generated by A and {c,} e Then oo : i= Proof: We shall estimate the terms hue gitl 3 by terms of a convergent serieSe Define D’ = { P, (8:8) |cos @cos f>r { for -i<ré<tl. Using the representation of the sphere in 3=space we can say that Dy consists of all points of the sphere whose x~coordinate is greater than r. Since all of the coordinate systems Cs give the same x-axis, D is independent of coordinate system. Now i . Jay {Ag (08) © NTF f < mil Mf dle, (eben Ay Qa L the right hand side of this inequality equals Tae. (Ales (9,4) ea*'q } . Multiplying each of these functions by cos 6 d8 and integrating gives By the definition of aA7** Minn") < Miah DT) sO") = (1-4) 2m, Defining 1 : 1 Ty tne r G, = A’ (\D') ar < 1-r) 20 dr = ‘ [ A "Vp ) ref r) 2n dr = du, we see Gs as a monotone increasing sequence bounded from above so that ao dG -,, ~G. S40 —- G, < wm. f=, (itl i 1 - 38 = . . ig ,i+l : : We shall bound Moa @ a by a fixed miltiple of G,,, - G,- For convenience in the following we use i= 0. Define oe = { IP, (ond) & ot 8 1 by = {lP,(08) ea} The corresponding function for A' =A, is A which has been C C8 previously defined. Then 1 a/2 _ ry) r G,-G, = | [ 2 Mi (Ag ge) ~My (agg) cos @ ae ar. on yor) = 9 [-a,7) so that /, (Ag gMDg) M, (Ag 9) = M, (ag) =, (4,1 Dg): Also if r> cos 6 then D6 is empty. Thus Now if r< = cos @, then D cos @ n/2 cos @ r | r (1) G,-G, -[" [ [Mag (Wg) - bh (AN Dg] dr cos @ d@, Since Ay is Steiner symmetric with respect to the coordinate system C, My (Age WG) = minl Ly (Ag,g)» 44 OG)) = minl{d; (Ag)» 2 are cos eos 6° Also g f so that the integrand is always non-negative and decreasing the range of integration to -w/2<@6< 0/2 and -cos 6 <r < cos 6 cos (M, (4g )/2) decreases the right hand side of (1). In this restricted range - 39 - ‘iM 15 )2 Jay (Ay ). With these considerations (1) implies yo ) cos 8 cos Le 5 orage [| os @ TH, (Kg) = 4 Bg (106)] cos @ dr ae (2) M, (Ag) w/2 cos @ cos —— = i i D, (0,8) ag dr cos 6 dQ, =r/2 J) cos 9 ° where D. is the characteristic function of A, Interchanging the order of integration LQ) /2 cos 8 cos 5 (3) Gc > / D, (0, ar ag cos 6 40. ° aa f2 Isle MM, ( ay cos @ cos f The integration in the variable r can now be performed to give 2 A | (4) G,=G, > / v D (0,0) cos*e[cos ani @) cos f] ag do, » Hie’ The nontrivial range of integration of g is that portion of Ag which is disjoint from Ag and the measure of this range is Hy (agli ,9) where Bog is the complement of Ange An immediate consequence of Theorem C (Appendix) is that wie (5) G,-G, 2 | / i cos 7alcos(M(ay )/2) = cos : 6) apg de we [4 (Ke) My bg Bog) Ae) (ne <ip|g 8. : -40- Performing the integration in the variable @ and using a trigonometric identity for the difference of sines gives | “se Oa (Ag) (6) oy-0, 2" (Ah ce? os MA’ cos 26 [ 0/2 — 9 bibl e , HMol oe de, Using the fact that for x > 0, =x < ~ sin x and a trigonometric identity for the difference of cosines we obtain from (6) [ve : Jy Bg Ace) ju, (A048...) sin ~1/2 five ce 4 (7 ) G, “G, 2 “aint Mibe? , Meee! 2 ] cos” 6 de, Since Mineo? [ine <4 ona HiSco! fio hee? a The trigonometric functions may be estimated from below by linear functions to yield (0/2 Ube NEL a,-0 2. [ Hie O46 2 > cos’ @d6 , on, -a/2 64 Multiplying the integrand by cos @ and applying Holder's inequality to the functions f£(@) = M, (ag M Ang)» g(8) =1 gives a” ant 3 Ga, 2 Ay Malla’, ~41- Since M (A) = fat), we have HM (M4) = Mak) = 5 ACA @a,) Thus Ww © 3 G.-G, 2 575 Ve (A A) . QebeDe Our next theorem is the analogue to Theorem 9. THEOREM 13. Let A be a measurable set on the sphere and {cs} be a periodic sequence of coordinate systems with a common origin. Then the Steiner sequence converges in measure to a set which is Steiner symmetric in all the coordinate systems. Proof: If A has a finite perimeter the argument is identical with the proof of Theorem 9. The second observation is that the class c= {alat converges in measure } is closed under complementation. For if A is the complement of A then xt is obtained from at by reflection through the yz plane. And if at converges then so does at, The third observation is that C is closed under monotone limits. For let Bo be a monotone sequence of sets of C. Then the BA converge to a set we shall denote by Bis Denote by Be the set to which the Steiner sequence B converges (n fixed). Since Steiner symmetrization takes a monotone family of sets into a family monotone to within measure zero, the sequence Be must be monotone up to measure zero and thus converges in measure to a set we shall denote by Bos Then e (1) L(B,OBe) < 1 (B,@B,) + KU (B® B.) + AA(B OB) ~42 = Since Steiner symmetrization takes monotone sets into monotone sets, it can be shown that ib (BLO a) < MB, © B) for all i. By choosing , 1 ~ . n such that & (BLO BL) < z & and jé (BY @B") < t & (both sequences converge) (1) becomes 4 (B® Be) $3 e+ (BLO BE) But B >B as i+, so that for all i sufficiently large jb (pt @ B®) < c. Oo. oa Thus B isin Cc. If we could show that CG was a ring we would be done. Unfortunately there is no direct proof of this and we must proceed differently. It is well known that the sphere is separable under the ordinary topology. And in fact circular discs with rational coordinates for their centers and rational radii form a basis, ! Since finite unions of circular discs have finite perimeter and every open set is the monotone increasing limit of such sets, every open set is in CG. Next every Gs. is in c.7? Lastly, since convergence is only to within measure zero, any set which differs from a Gs by a set of measure zero is in C. But this includes all measurable sets. Qe EeDe The next theorem is the analogue of Theorem 4 on the sphere. THEOREM 14. Let F; G be integrable functions on the sphere and 4H a 2 be monotone decreasing on [O a]. Let F.,G _ be the rearrangements 22) See ref. 12, p. 3 for definitions. -143- of F, G based on the sets DY = { P(@,f)|cos @ cos J >r } » Then [fre G(Q) #(Q (P,Q)) a4P dQ f/f F'(P) G'(Q) H(@(P,Q)) aP aa Proof: As on several preceding occasions the general case follows from the case when F, G are the characteristic functions of measurable sets, A and B. Let Cys C. be two coordinate systems such that the angle between the arcs {P, (8,4)|f = o} and { P,(65)19 = of at their point of intersection, P, (0,0) = P. (050) is an irrational multiple of a. Now Lim at, Lim pt are Steiner symmetric in both coordinate systems and thus must be circular dises (the pD'ts).- An application of Theorem 10 completes the proof. 23) Ref. 8, pe 90. APPENDIX Theorems A, By, C are simple measure theoretic results which are used in the main part of the text. THEOREM A. If Ay B are integrable sets on G then M fal4(a OB) >Oo}=0 if and only if (A) = 0 or (8) = 0. Proof: jh (A C1 2B) - [Dw D (27x) ax and fu (A(\ 2B) dz [Povo Byo0 ax dz [Qe [Oye%r dz dx -[Qep (Bu!) dx =A (a) 40 (8) ‘ Since Va. (A (\2B) is non-negative, as (A) ji (pv!) is zero if and only ii if Af 2[c4(a 2B) > 0} = 0. Lastly, /A (pv!) = / D(z) A(z!) dz. Since D, (2) is non~negative and A(z’) is positive everywhere, i (pv!) is zero if and only if as (B) = 0. The theorem then follows. THEOREM B. Let A, By C be integrable sets on G with JAA) < JAB) sf (Cc). Then there exists al, c! such that fea) =A (A), LA Mel) = fL(C) and alcscol, Proof: We will construct a’, The modification for c! is obvious. If A’ or B have zero measure we are through. Otherwise consider Mian ata} - There exists z, such that JA (a 2518) >} sup K (A 27's) . ZeG Define A, =(2 ANB B, = B-a,,A =a-27'a Then A, CB 1 1 an 74 171° t=” * and £U (Ay) + (a4) = LA(A). Inductively we choose 244 such that / oe By 23 ZeG + i and define . * * Aggy = Ay Ulay,, 4p 08, eL . Bist = Bm aay * Asay = aT Nash By It is easily shown by induction that A, CB %. (1) J (Ag, 4) = M4 (Ay) + Utay ay Ta B,]s * (2) ft (Ag) + Cag) = (ADs and { Af is a monotone increasing sequence and { By } : fart are monotone decreasing. Thus -~ 46 = * 4 exist. * * * At=limA,, B_=limB,., and A =limA a . 2 oOo . a eas] a Rea jc Also KB a Zz "yD decreases monotonely to j (aie B®). Farther more, since A is monotone and the (A,) < (B) < «, we have i Kil = # ~~} _* A (a 28g) = gay) = Ag) = A (Aggy A) 70 Thus sup Man 27's”) < sup (a, N2'a™) <2 Jag 25 Bs) > 0 so that zeG ZEG * A -1.*, = . * * Manz B.) = 0. By Theorem A either jb (A,) or MB, equals zero. * 1 Now B =B-A_ and ce (3) PE (Be) = (1(8) = fA (4!), Teking the limit of (2) as io gives (4) Har) + AA (a') =fA(a) That A is the one with zero measure follows from (3) and (4). Then from (4) it follows that AA (a') =H (A). Since A, CB for all i, alos. Q.E.D COROLLARY. If A is integrable then there exists al such that 1 BA = aa!) and the closure of A’ is compact. Proof: If G is compact then any closed subsets are compact and the corollary is proved. If G is not compact, since it is locally compact, there exist compact sets of arbitrarily large measure and thus one whose measure is greater than ié (A). The corollary then follows from the theoren. -147 = THEOREM C. Let F(x) be a positive integrable function on G and let A be an integrable set. Then there exists a real number t such that M {xl F(x) > tf 2 f(A) 2 4 {xl F(x) > t} DEF F, | #00 dx < [ F(x) dx +t [4(A) - A(R), t and Furthermore if equality holds then F 4 SA {xl F(x) > +} except for a set of measure zero. Proofs Let t = ant {t|.(F,) < AX(a)} fo) The sets F 4 are monotone increasing as +t decreases and R= Lim Fy +5° 20+ Thus KM (F,) is continuous on the right and (FS UG) is] eo Also { xlF() > tof = a Fy -1/n 80 that n= o M{xlF) 2+.) = 1am UR 4) 2 (A). 2 ° Secondly, let A, = A, Ay = AME fx] F(x) > t,4 - As], ana 2 = Age Then J re) = [ rte) ax + ff wte) ax +f 2x) ax A A A A 1 2 3 (Ar A) =A By Theorem B there exists a subset, C, of AU Ag whose measure is =~ 43 = /4 (F) “Haye Since F(x) >t, on F, and F(x) < t, on C we have [0 dx < [ F(x) dx “f F(x) dx ff F(x) dx A F ; Ana, Cc Awa, NC t ° with strict inequality holding unless 4 (Cc) = 0, that is, unless F, CA t to within measure zero. A similar construction replaces Ag - Ay Nc by a subset of {x| F(x) = toh - (A, - AO C) and a corresponding strict inequality results unless (A, - A, ‘\C) = 0. Combining the two steps yields rte) ax gf r(a) ax + tg A) -A(F, )] | 0 x 2, (x) dx ot Hl [ey 9 with strict inequality holding unless Fy CAC }x|F(x) > tof except for sets of measure zero. LEMMA TO THEOREM 11. Let I be a rectifiable closed curve on the sphere given by { (e(s)s O(s))|o<s<s } where s is the are length from the point (@(0), 6(0)) along the positive orientation of the curve. For ~t/2<a<b < 0/2, define I= {sla < @(s) <b or @(s) =a and 250 or @(s)}=b and e <0} . = AQ = Statement of Lemma to Theorem 11. If [ is bounded away from P(O,-7) and the winding number of every point not on I is either zero or one and F(e) = Liu, { AP (9.0) is on I or has winding number 1 then f(s) and @(s) are absolutely continuous functions of s on I and ain) ~ et s fie sl ds Proof: We fist show absolute continuity. Choose § > 0 such that —/2+§ <a and b<20/2~%8. Thus Tv TC [sl- $+8 < os) <5 +8h. This last set is open and thus consists of a countable union of disjoint open intervals. On each of these intervals we have if Sy > 8, and each are in the interval 14 x V (a9, )* + cos*[o(e,)} (46, )* 8, -s. = lim A + cos c.)} (6, 1 9 |aino £4 uo where gf is a partition of [s oye 8, By = <0, SH < gy = By and AS, 5 ap, are the differences of consecutive values of the correspond=- ing functions, |g] = max X; “ X,_,¢ It follows that i 8,7 38,2 Lin 2,1 48,1 2 > |a(s,) - a(s,)| and 8,7 8,2 9 2 tin 2 cosle(e; 1 [46,1 2 cos - 8)| Als.) - Als.) Ley Now let qy> i, be a sequence of intervals [a, »b,] contained in {sl- W/2 + & < @(s) < 0/2 - 8} such that 2b, “a, <esinS. Then dy (by ~ a) i=1 < & sin > =¢ cos(e = 8) cos(Z-5) 2 2 2 [O(b,) = Ola, II < Therefore @(s) is absolutely continuous. A similer computation shows @(s) to be absolutely continuous and in fact by letting § 70 the absolute continuity of @(s) on the whole interval [0,8] can be shown dg . Next we will prove 2|F(b) - F(a)| < [ss ds» Let I Do) = 1 if P(a,f) is on [ or has winding number 1, and D (0) = 0 otherwise. Similarly for QD (g). Then alate) = xta)l = 1f" (6) - Dion asl = </ , ID w) - 6 | dy , “1 If we can show the last integral to be less than or equal to [ [Mas I the proof will be complete, To do this we need two lemmas. LEMMA 1. Let N,(y) be the number of solutions to the equation §(s) = y, sé. Then a _ ag [. Ny) ay = i sal ds . LEMMA 2. ID wy) - Diy)! < NL @) except for a set of y of measure Zerde w That [- ID) - Dw dy < Fl ds is an immediate consequence of these two lemmas. Proof of Lemma 1. Consider the class, C, of all sets aA for which a [C yoae [hes If A is an interval then [ 32) ds = Var pg. But a result of S. A A 2 Banach shows that intervals are in C. Next, if A, WU Ay = A. and 3 A,(\A, is empty then N, (y) +N, (y) =N, (y), so that 1 2 A, Ay Ag N )a N ) dy = N, ay, Jue r+ fag y [re y | ag a / ag g -[ gf . I. gla J, Waal ee= J, lal ae 3 The integral equalities are to be interpreted as asserting also that if any two of the integrals exist then the third does. From this it follows that CG is closed under relative complements and finite unions of disjoint members of C. If { ay} is a monotone sequence of members of C¢ converging to A, then Ny (y) converges monotonely to N,(y)s i T T / Ny (y) dy converges monotonely to / N, (y) dy “rt i = and f FA ds converges monotonely to [ Fed ds . A, 8 aS 24) Ref. 13, Theorem on page 48. 25) Ref. 14, Theorem 2 page, 228. ~52= Thus C is closed under monotone limits. In particular since every open set is the countable union of disjoint opern intervals, every open subset and every closed subset of { s|-n/2 +§ < @(s) < 07/2 = 3{ are in GC. Finally the countable intersections of open sets (G § ts) and countable unions of closed set (Fg's) belong to ©. Now if A is measureable there exists Fe » Gg such that Ey CAC Gy and ji (Gs ~ Fy) = 0. 2 We have and d a . [a ds = [ [Blas provided G, C {s|-n/2 +5 < @(s) < 0/2 “sh, 8 The last statement implies I Np 7) dy = fi No 7) dy. Since Nps Noy 9 this implies N and N and thus also N, must differ on a set of Fy Gs A measure zero. Thus /N,(y) dy exists 198) ds = jad) ds. A F ds A ds Thereforeevery measureable set is in © and in particular I isin C. QeHe De Proof of Lemma 2. If Oy) and DW) are both 1 or both 0, it is trivial that IOo~) - Owl < Noy) « Also if there exists s such that a< @(s) <b and f(s) =y then ID) - Dy)! <isnyy). That leaves the case where exactly one of the points P(bsy) and P(a,y) has winding number 0 and no point of { P(8.y)la <@< vo} are on the curve T. Suppose (b,y) is the point with winding number 0. Then, by continuity, all of the points { F(9,y)la <9< bf must have winding number 0 and (a,y) must be a point on I. We shall now have to exclude sets of meagure gero. First let $$ > 0, “9/2 +3<a<b<a/2+5 and A ww a ‘ Ze. A= {sl-3+8< os) <3 -sf. Then N, Oy) is integrable by the preceding lemma and therefore finite almost everywhere. We exclude those y for which ni) is infinite. Secondly let 84s Sos eee » 8 be the solutions to O(s)=y, sede Since #(s) and @(s) are absolutely continuous on A, df/ds and de/ds exist and are finite almost everywhere. We exclude that set of y for which there exists s such that Q(s) =y, sc A and either daf/ds or d@/ds do not exist or are not finite. Finally we look at those S; for which o(s, }=a. It is claimed that _ ae - KX; { s|e(s) =a and as F 0} = 0. The reason is as follows de _ @(s_+ A) = 8(s) . de ds ~ © * 4 < as +E for all |al sufficiently small. Thus if d@/ds #0 and @(s)= a then @(s')= a can have no other solution in some neighborhood of s. Thus no point of {sle(s) =a and d6/ds # of can be a density point of this set. It follows that this set is of measure zero.”* The correspond= ing set of y must also be of measure zero and will be excluded. 27) Ref. 15, ppe 123-124, ist edition; pp. 185~187, 2nd edition. Now let us examine the set of y remaining. For each y in this set we have: There exists s with @(s) = y, 6(s) = a. There are only a finite number of solutions to @(s) = ys, -9/2 +§ < @(s) < 1/2 -§ For each solution to $(s) = y, 6(s) =a we have df/ds exists and d@/ds = 0. we Since 1 = ye)? + cos” @ (SB) : ae cannot be zero and in fact ag = ds ~cos@° Let 815 Sos see » S, be the solutions to @(s) =y, 9(s) =a. Since there are only a finite number of solutions to O(s) = yy, -0/2 + § < Os) < <u/2- §, there is anarc { P(@y)la-ce<@<a +e} such that the only point in common with [ is (as,y). For each S59 L= 1,5 oes 9 such that d@/ds > 0 there exists an E; such that @(s) <y for 8 & (s, — Es s,) and @(s)>y for se (S55 8, + E; )s Similarly for df/ds < 0. Thus there is a crossing of the arc of [. If we look at the stereographic image of the arc and of [ on the yz plane using the point P(O,-1) to project from, we see that if ag/ds > 0. the contribution to the winding number of (a - €,y) minus the winding number of (a + sy) is +1, while if df/ds <0 the contribution is <1. Letting W(@,0) be the winding number of the point P(@,f) Wa -84y) = Wa +8.y) = 2 sen 2 i i °* Bat Wla+égsy) = 0 and Wla-g§ .y) is non-negative so that ~ sgn ag > 0 i ds|s, 7 . i = 55 = Since there is at least one term in the sum, there must exist one positive term so that Ni(y) 2 12 O.(y) - 6.(y)|. A similar argument is valid if 6, (y) = 1 and 6) = 0 and completes the proof. 1. 26 Be he De Ge Te 8. Jeo 10. 11. 126 Ae = 56 = REFERENCES Haar, Der Massbegriff in der Theorie der Kontinuierlichen Gruppen, Annals of Mathematics, 345 147-169 (1933). Andre Weil, Ltintégration dans les Groupes Topologiques, Paris (1953). Fe Re Ge Le We Je Te We Ge Riesz, Sur une Inégalité Intégrale, Journal London Math. Soc. 5,4 162-168 (1930). Me Gabriel, The Rearrangement of Positive Fourier Coefficients, Proc. London Math. Soc. (2) 33 (1932). H. Hardy, Je E. Littlewood, G. Polya, Inequalities, Cambridge University Press (1952). H. Loomis, An Introduction to Abstract Harmonic Analysis, Van Nostrand (1953). 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