APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS

APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS 1. Itroductio Let C be a bouded, covex subset of. Thus, by defiitio, with every two poits i the set, the lie segmet coectig these two poits is also cotaied i the set. A good way is to thik of covex sets as the itersectio of a family of closed half spaces. Ideed C = {H : C H, H closed half space} =: C H. Evidetly, C C H because C H is the itersectio of closed sets all of them cotaiig C. To see that C = C H assume otherwise, i.e., assume there exists a C H but a / C. Sice C is closed ad covex we kow, by the Hah Baach theorem, that there exists a plae that separates C from a. I other words there is a closed half space that cotais C but ot a. Thus, a / C H cotrary to our assumptio. I order to avoid degeerate sets, we shall assume that the set is closed ad its ope iterior is ot empty. Ay covex set is Borel measurable ad hece has a volume. Cosider the set of all closed ellpsoids E that are subsets of C. Sice the iterior of C is ot empty the set C cotais at least oe such ellipsoid. ecall that ellipsoid are give as the set of all x so that (x b), A(x b) 1. Here, A is a positive defiite matrix, b is the ceter of the ellipsoid ad, deotes the stadard Euclidea ier product. We ca write this expressio i a alterative way by usig the orthoormal set of eigevectors of A, e j as a basis ad obtai (x b), e j 1. (1) λ 2 j Note that λ j is precisely the legth of the j-th semi-axis of the ellipsoid. Thus, the coditio that E is a subset of C meas that ay x that satisfies (1) is also i C. The volume of the ellipsoid ca be readily calculated to be vol(e) = S 1 λ j, where S 1 is the surface area of the Euclidea uit sphere. Or expressed i terms of the matrix A vol(e) = S 1 det(a 1/2 ) = S 1 (det(a)) 1/2. Amog those ellipsoids cotaied i C, there exists oe of largest volume. Sice, C is bouded, it is a subset of a large ball B of radius ad ellipsoid which is a subset of C is also a subset of B. From this we see that the set D := {(A, b) : (x b), A(x b) 1 implies x C} Date: April 24, 2010. 1

2 APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS is a bouded subset of (+3) 2. Moreover, the complemet of D i (+3) 2 is ope. Ay ellipsoid that has a o-empty itersectio with the the complemet of C ca be perturbed i ay directio without losig that property. Thus, the set D is compact ad sice the volume is a cotiuous fuctio, there exists a ellipsoid of largest volume. With these prelimiaries we ca ow formulate the volume ratio problem for covex sets. Poblem 1.1. Amog all covex sets C fid the oe that maximizes the volume ratio vol(c) vol(e(c)) where E(C) is the ellipsoid cotaied i C with largest volume. Ituitively, the set C should be rather slim i order to keep the volume of the ellipsoid small but at the same time makes large cotributios to the volume that is ot covered by the ellipsoid. By shiftig coordiates, we may assume that the ellipsoid of maximal volume has its ceter at the origi. It is a stadard result from liear algebra that ay symmetric matrix A ca be writte i the form A = S T S. Thus, by mappig C oto the covex set {Sx : x C} we may assume that the ellipsoid of maximal volume is i fact a ball, ad sice scalig does ot chage the volume ratio we may assume that the set C has the Euclidea ball as its ellipsoid of maximal volume. I these terms, the aswer to the above problem ca be stated very simply as Theorem 1.2 (Keith Ball). Amog all covex bodies whose ellipsoid of maximal volume is the Euclidea ball, the maximum of the volume ratio is attaied by the simplex, i.e., regular triagle i two dimesios, tetrahedro i three dimesios, etc. We shall ot prove this theorem, but istead solve this problem for symmetric covex sets. ecall, that a covex set C is symmetric with respect to the origi if with x C we also have that x C. Now we ca formulate a correspodig theorem. Theorem 1.3 (Keith Ball). Amog all symmetric covex bodies whose ellipsoid of maximal volume is the Euclidea ball, the maximum of the volume ratio is attaied by the cube. The proof will proceed via Brascamp-Lieb iequalities, bit first we have to characterize the largest ellipsoid. Discuss Examples via pictures 2. Fritz Joh s Theorem We have see that i ay covex set a ellipsoid of largest volume exists. The followig theorem of Fritz Joh is a characterizatio of it. Theorem 2.1. Assume that the uit ball is the a ellipsoid of largest volume of a covex set C. The there exist touchig (or cotact) poits u j, j = 1,..., m < ad positive umbers c j, j = 1,..., m so that c j u j = 0 (2) ad c j u j u j = I. (3)

APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS 3 Coversely, if the uit ball touches a covex set i poits u j, j = 1,..., m for which there exist positive umbers c j, j = 1,..., m so that both (2) ad (3) hold, the the uit ball is the ellipsoid of largest volume cotaied i C. Explai meaig via pictures emark 2.2. Note by takig the trace we lear that c j = ad hece I ca be writte as a covex combiatio of rak oe projectios of the form u j u j where u j is a cotact poit. There ca be a ifiite set of cotact poit, e.g., take for C the uit disk. Thus, i priciple oe might thik that m may be ifiity. I our applicatio, however, we will see that m ca be chose to be fiite. Note that the cotact poits i (2) ad i (3) may ot be uique. emark 2.3. Joh s theorem implies that if the Euclidea uit ball is the ellipsoid of maximal volume, the the set C is cotaied i the Euclidea ball cetered at the origi with radius. To see this we observe that C is a subset of the closed half space u i, x = 1 for i = 1,..., m. Hece x 2 = c j u j, x 2 c j =. j Proof. We follow here very closely Keith Ball. Here is a first observatio. Cosider the uit ball ad deform it slightly ito a ellipsoid. This amouts to cosider the quadratic form I + δh where H is a symmetric matrix, ad δ is some small umber. Likewise, the ceter of the ball will be also shifted ad hece we cosider the quadratic form Expadig i δ we get Q δ (x, x); = (x + δ 2 h), [I + δh](x + δ h). (4) 2 Q δ (x, x) = x, x + δ [ h, x + x, Hx ] + O(δ 2 ). (5) The term proportioal to δ ca be writte as [ ] Tr H T (x x, x). where H = (H, h). Note that for ay two ( + 1) matrices A, B Tr [ A T B ] is a ier product ad hece turs the ( + 1) matrices ito a ier product space. Cosider the set of matrices (u u, u)

4 APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS where u is a cotact poit. Now we cosider the set of all covex combiatio ad take the closure if ecessary. I this way we get a closed covex set G of ( + 1) matrices. I fact, this set is bouded, sice for aym = (A, a) G x, Ax j x, u j 2 x 2 p j = x 2, ad x, a x p j = x, for all x. Thus, G is a compact covex set. We wat to show that the matrix ( ) I, 0 is i this covex( set. Suppose ) ot, the there exists a liear fuctioal o the space of matrices that separates I, 0 from G. That is, there exists a matrix H = (H, h) with H T = H ad a costat κ such that [ ( )] Tr H T I, 0 > κ (6) ad [ ] Tr H T M κ ε < κ for all M G. Sice all matrices i M are of the form (A, b) where A has trace oe, we ca replace (H, h) by ( H 1 Tr(H), h) ad the above two iequalities are replaced by [ ( )] Tr H T I, 0 = 0 (7) ad [ ] Tr H T M ε < 0 Now cosider the ellipsoid defied i (4). For ay cotact poit u we have usig (5) [ ] Q δ (u, u) = 1 + δtr H T (u u, u) + O(δ 2 ) 1 δε + O(δ 2 ) > 1 (8) for δ small, egative. Thus, the cotact poits are outside the ew ellipsoid. Sice the boudary C is a compact set ad the fuctio x Q δ (x, x) is cotiuous, we fid that the ellipsoid defied be Q δ (x, x) 1 is a set strictly cotaied i C for δ sufficietly small. Now, the volume of the ellipsoid is S 1 det (I + δh) = S 1 1/2 (1 + δµ j) 1/2 where µ j are the eigevalues of H. The umbers 1 + δµ j > 0, j = 1,..., ad j µ j = 0. By the arithmetic-geometric mea iequality ( (1 + δµ j ) (1 + δµ ) j) = 1

APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS 5 with equality oly if all the umbers 1 + δµ j are the same. Sice the trace of H vaishes, this is the case if all the µ j s vaish ad hece H is the zero matrix. Thus, if H is ot the zero matrix, we are doe. So we assume that it is the zero matrix. I this case, the ew ellipsoid is the Euclidea ball traslated by δ h ad strictly cotaied i C. Makig the radius slightly 2 larger we still have ( a ball ) i C with a larger volume, which is a cotradictio ad proves the claim that I, 0 G. By Caratheodory s theorem there exist at most dimesio of uderlyig space +1 poits, i.e., (u j u j, u j ), j = 1,..., m ad umbers p j > 0, p j = 1 with such that m ( ) I, 0 = ( + 3) 2 + 1. p j (u j u j, u j ). This proves the hard part of Joh s theorem. Now we show the uiqueess of the ellipsoid. Let E be he ellipsoid of largest volume, E = {x : (x a), A(x a) 1} for some vector a C ad some positive defiite matrix A. The plae P i give by u i, x = 1 maybe touches E but certaily does ot cut it. Thus, we kow that for some 0 < b i 1 the plae u i, x = b i touches E at some poit x i, i.e., ad u i, x i = b i (x i a), A(x i a) = 1. Hece u i must be proportioal to the gradiet of x, Ax at that poit ad therefore u i = A(x i a) A(x i a). Sice x i is o the boudary of the ellipsoid we have that Hece, Now, b i = u i, x i = u i, (x i a) + u i, a = (x i a), A(x i a) A(x i a) u i, A 1/2 u i = A(x i a), A 1/2 (x i a) A(x i a) 2 + u i, a = 1 A(x i a) = b i u i, a. 1 A(x i a) + u i, a A(x i a) A 1/2 (x i a) A(x i a) 2 = 1 A(x i a) = b i u i, a

6 APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS ad usig (2) ad (3) we lear that or TrA 1/2 = j c j u i, A 1 u i TrA 1/2 c j (b i u i, a ) 1. If we deote the eigevalues of A 1/2 by µ j, j = 1,...,, the volume of E is proprtioal to ( ) TrA 1/2 µ j 1. j Thus, sice E is, by defiitio the ellipsoid i C of maximal volume, j µ j = 1 ad all the eigevalues must be equal to 1. Hece A = I what we had to show. 3. Brascamp-Lieb Iequality i the versio of Keith Ball Cosider for ay give vectors a j, j = 1,..., m the expressio m f j( a j, x )dx m f j pj. I order that the maximizatio problem makes ay sese, we eed to have that 1 =. p j Now the supremum over all fuctios f j might be fiite or ot. I ay case it was show by Brascamp ad Lieb i the seveties that it suffices to optimize the ratio over Gaussia fuctios, i.e., fuctios of the form f j (x) = e s jx 2. While this simplifies the maximizatio problem greatly, the determiatio of the umbers s j is a formidable o-liear problem ad depeds very much o the vectors a j. For geeral vectors, othig is kow about the costat, except whe it is fiite. It was observed by Keith Ball that the Brascamp-Lieb result greatly simplifies if the vectors a j satisfy the Fritz Joh coditio. I order to be i agreemet with the otatio i the previous sectio we shall deote these vectors by u j ad assume that there exist positive costats c j so that c j u j u j = I. ecall, by takig the trace i above matrices, that c j =.

APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS 7 Theorem 3.1 (Brascamp-Lieb, Keith Ball). If the vectors u j, j = 1,..., m satisfy the Fritz Joh coditio the for o-egative fuctio f j, j = 1,..., m ( f j ( u j, x ) c j dx f j (x)dx). (9) The costat oe is sharp. emark 3.2. The cases of equality are istructive. Pick f j (x) = e x2, j = 1,..., m the we see that the left side is e c j u j,x 2 dx = dx while the right side is ( e x2 dx ) P m c j = e x 2 e x 2 If m = ad the vectors u j form a orthoormal basis, ad c j = 1, j = 1,...,, the there is equality for ay collectio of f j s. Proof. We will prove this theorem usig the heatkerel method developed by Carle-Lieb-Loss, as well as Beett-Carbery-Christ-Tao. It suffices to show the iequality for ice fuctios, o-egative, with compact support. The fudametal solutio of the heat equatio, the heat kerel, u t = u i is give by G(x y, t) = (4πt) 2 e x y 2 /4t dx. Now we apply the heat kerel to the fuctios f j ( u j, x ) ad ote that G(x y, t)f j ( u j, y )dy = f j ( u j, x, t), i.e., it is a fuctio of the variable u j, x oly. The fuctio f j (y, t) is give by f(y, t) = (4πt) 1 2 e (y z)2 /4t f j (z)dz. Obviously, f(y, t)dy = f(y)dy, i.e., it does ot chage with time. Thus, if we replace i (9) the fuctios f j ( ) by the fuctios f j (, t) the right side of the iequality does ot chage. We shall show that the left side icreases with t ad compute the limit. For t > 0 it is obvious that the fuctios f j ( u j, x, t) are smooth fuctios of their variables. Moreover, sice the f ( ) are o-egative, the fuctios f j (, t) are strictly positive for t > 0. Hece we may calculate d dx dt

8 APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS = c k k=1 j k dxf c k 1 k ( u k, x, t) (f k ( u k, x, t)) Next we itegrate by parts. There are o boudary terms, sice the fuctios all decay fast at ifiity. Please ote, that c k 1 may coceivably be egative, however, the combiatio f c k 1 k ( u k, x, t) (f k ( u k, x, t) still has fast decay at ifiity. Hece we get that = k,l=1,k l = + k=1 c k c l j k d dt dx [ m c k f j ( u j, x, t) c j dxf c k 1 k ( u k, x, t) j k,l c k (1 c k ) k=1 f c k 1 k j k ] (f k ( u k, x, t)) ( u k, x, t)f c l 1 ( u l, x, t)f k( u k, x, t)f l( u l, x, t)u k u l l f c k 2 k ( u k, x, t)f k( u k, x, t) 2 u k 2. As metioed before, for t > 0 the fuctios f j (, t) are all strictly positive ad it is coveiet to write f j (y, t) = e αj(y,t) ad dx = dµ(t). I this otatio, = d dt [ m α k( u k, x, t) 2 c k u k 2 k=1 dx ] α k( u k, x, t)α l( u l, x, t)c k c l u k u l dµ(t). (10) k,l=1 Now we shall show that the expressio i [ ] is oegative which amouts to showig that the symmetric matrix with elemets ci δ i,j c i u i u j c j = c i [ δi,j c i u i u j ] defies a o-egative quadratic form. This follows, oce we show that the symmetric matrix with elemets ci u i u j has eigevalues less tha oe. Let (y 1,..., y m ) be a eigevector. Thus ( m ) ci u i u j y j = λy i i = 1,..., m. (11) i=1 We may assume that the vector i x := u j y j 0 i=1

APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS 9 for otherwise λ = 0. Multiplyig (11) by c i u i ad summig over i yields c i u i u i x = λx i=1 ad sice m i=1 c iu i u i = I, λ = 1. hece the eigevalues are zero or oe ad the expressio (10) is o-egative. Thus the left side of (9) is a o decreasig fuctio alog the heat flow, while the right side stays of (9) stays put. Now we calculate lim dx. t A simple calculatio yields 1 dx = (4πt) 2 ad by scalig x 4πtx, = e π x 2 ( P m e c j u j,x 2 4t ( e π u j,x z t e z2 4t fj (z)dz) dx e u j,x z 2t e z2 4t fj (z)dz) dx A simple domiated covergece argumet shows that the limit as t is give by ( ( ) e π x 2 f j (z)dz) dx = f j (z)dz which proves the iequality. The proof of Theorem 1.3 is ow straightforward. We may assume that the set C has a Joh ellipsoid that is the Euclidea uit ball. Hece the set C is a subset of the set C H geerated by the itersectio of the half spaces u j, x 1, where u j are the vectors give i Fritz Joh s theorem. Sice we u j, u j is also amog those vectors we kow that C H is the give by the itersectio of the slabs The volume of this set is give by vol(c H ) = 1 u i, x 1. χ( u i, x )dx where χ is the characteristic fuctio of the iterval [ 1, 1]. By Theorem 3.1 we have that vol(c H ) = χ( u i, x ) c jdx ( χ(z)dz) c j = 2 P m c j = 2. Hece, vol(c) vol(e) 2 vol(b )

10 APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS where B, the uit ball, is the ellipsoid of largest volume cotaied i the cube of side legth 2.