The isotropic position and the reverse Santaló inequality
Apostolos Giannopoulos, Grigoris Paouris, Beatrice-Helen Vritsiou
aa r X i v : . [ m a t h . M G ] D ec The isotropic position and the reverseSantal´o inequality
A. Giannopoulos, G. Paouris and B-H. Vritsiou
Abstract
We present proofs of the reverse Santal´o inequality, the existence of M -ellipsoids and the reverse Brunn–Minkowski inequality, using purely convexgeometric tools. Our approach is based on properties of the isotropic posi-tion. We work in R n , which is equipped with a Euclidean structure h· , ·i . We denote thecorresponding Euclidean norm by k · k , and write B n for the Euclidean unit ball,and S n − for the unit sphere. Volume is denoted by | · | .A convex body K in R n is a compact convex subset of R n with non-emptyinterior. We say that K is symmetric if x ∈ K implies that − x ∈ K . We saythat K is centered if its barycenter is at the origin, i.e. R K h x, θ i dx = 0 for every θ ∈ S n − . For every interior point x of K , we define the polar body ( K − x ) ◦ of K with respect to x as follows:(1.1) ( K − x ) ◦ := { y ∈ R n : h z − x, y i z ∈ K } . Note that ( K − x ) ◦◦ = K − x .The purpose of this article is to present an alternative route to some funda-mental theorems of the asymptotic theory of convex bodies: the reverse Santal´oinequality, the existence of M -ellipsoids and the reverse Brunn–Minkowski inequal-ity. The starting point for our approach is the isotropic position of a convex body,which can be shown to simultaneously be an M -position for the body if its isotropicconstant is bounded. The new ingredient in this paper is a way to also show, usingonly basic tools from the theory of convex bodies and log-concave measures, thatevery convex body with bounded isotropic constant satisfies the reverse Santal´oinequality, and then that all bodies do.We first recall the statements and the history of the results. The classicalBlaschke-Santal´o inequality states that for every symmetric convex body K in R n ,the volume product s ( K ) := | K || K ◦ | is less than or equal to the volume product ( B n ), and equality holds if and only if K is an ellipsoid. More generally, for everyconvex body K , there exists a unique point z in the interior of K such that(1.2) | ( K − z ) ◦ | = inf x ∈ int( K ) | ( K − x ) ◦ | , and for this point we have(1.3) | K || ( K − z ) ◦ | s ( B n )(with equality again if and only if K is an ellipsoid). This unique point is usuallycalled the Santal´o point of K and is characterized by the following property: thepolar body ( K − z ) ◦ of K with respect to the point z has its barycenter at the originif and only if z is the Santal´o point of K . Observe now that the body K − bar( K )is centered and it is the polar body of ( K − bar( K )) ◦ with respect to the origin,hence 0 is the Santal´o point of ( K − bar( K )) ◦ . This means that for every centeredconvex body K ,(1.4) s ( K ) = | K || K ◦ | = inf x ∈ int( K ◦ ) | K ◦ || ( K ◦ − x ) ◦ | , and this allows us to restate the Blaschke-Santal´o inequality in a more concise way:for every centered convex body K in R n , s ( K ) s ( B n ), with equality if and onlyif K is an ellipsoid.In the opposite direction, a well-known conjecture of Mahler states that s ( K ) > n /n ! for every symmetric convex body K , and that s ( K ) > ( n + 1) n +1 / ( n !) in thenot necessarily symmetric case. This has been verified for some classes of bodies,e.g. zonoids and 1-unconditional bodies (see [28], [18], [30] and [10]). The reverseSantal´o inequality, or the Bourgain–Milman inequality, tells us that there exists anabsolute constant c > (cid:18) s ( K ) s ( B n ) (cid:19) /n > c for every convex body K in R n which contains 0 in its interior. The inequality wasfirst proved in [5] and answers the question of Mahler in the asymptotic sense: forevery centered convex body K in R n , the affine invariant s ( K ) /n is of the orderof 1 /n . A few other proofs have appeared (see [20], [15], [25]), the most recent ofwhich give the best lower bounds for the constant c and exploit tools from quitediverse areas: Kuperberg in [15] shows that in the symmetric case we have c > / c > π /
32. It should alsobe mentioned that Kuperberg had previously given an elementary proof [14] of theweaker lower bound s ( K ) /n > c/ ( n log n ).The original proof of the reverse Santal´o inequality in [5] employed a dimensiondescending procedure which was based on Milman’s quotient of subspace theorem.Thus, an essential tool was the M M ∗ -estimate which follows from Pisier’s inequal-ity for the norm of the Rademacher projection. In [20], Milman offered a second pproach, which introduced an “isomorphic symmetrization” technique. This isa symmetrization scheme which is in many ways different from the classical sym-metrizations. In each step, none of the natural parameters of the body is beingpreserved, but the ones which are of interest remain under control. The M M ∗ -estimate is again crucial for the proof.Our approach is based on properties of the isotropic position of a convex bodyand combines a very simple one-step isomorphic symmetrization argument (which isreminiscent of [20]) with the method of convex perturbations that Klartag inventedin [12] for his solution to the isomorphic slicing problem. Aside from the use ofthe latter, the approach is elementary, in the sense that it uses only standardtools from convex geometry; namely, some classical consequences of the Brunn–Minkowski inequality. Recall that a convex body K in R n is called isotropic if ithas volume 1, it is centered and its inertia matrix is a multiple of the identity: thereexists a constant L K > Z K h x, θ i dx = L K for every θ ∈ S n − . It is relatively easy to show that every convex body has anisotropic position and that this position is well-defined (by this we mean uniqueup to orthogonal transformations): if K is a centered convex body, then any linearimage ˜ K of K which has volume 1 and satisfies(1.7) Z ˜ K k x k dx = inf nZ T ( ˜ K ) k x k dx : T is linear and volume-preserving o is an isotropic image of K . This also implies that any isotropic image of K hasthe same isotropic constant, and thus L K can be defined for the entire affine classof K . One of the main problems in the asymptotic theory of convex bodies is thehyperplane conjecture, which, in an equivalent formulation, says that there existsan absolute constant C > L n := max { L K : K is isotropic in R n } C. A classical reference on the subject is the paper of Milman and Pajor [21] (seealso [7]). The problem remains open: Bourgain [4] has obtained the upper bound L K c √ n log n , and Klartag [12] has improved that to L K c √ n – see also [13].However, in this paper we only need a few basic results from the theory of isotropicconvex bodies and, more generally, of isotropic log-concave probability measures.All this background information is given in Section 2; there we also list a few morenecessary tools from the general asymptotic theory of convex bodies and, in orderto stress the fact that all of them are of purely “convex geometric nature”, weinclude a short description of the arguments leading to them.In Section 3 we prove the reverse Santal´o inequality in two stages. First, usingelementary covering estimates, we prove a version of it which involves the isotropicconstant L K of K . heorem 1.1. Let K be a convex body in R n which contains in its interior.Then (1.9) 4 ns ( K ) /n > ns ( K − K ) /n > c L K , where c > is an absolute constant. Then, we use Klartag’s ideas from [12] to show that every symmetric convexbody K is “close” to a convex body T with isotropic constant L T bounded by1 / p ns ( K ) /n . Theorem 1.2.
Let K be a symmetric convex body in R n . There exists a convexbody T in R n such that (i) c K ⊆ T − T ⊆ c K and (ii) L T c / p ns ( K ) /n , where c , c , c > are absolute constants. Since K and T − T have bounded geometric distance, we easily check that s ( K ) /n ≃ s ( T − T ) /n . Then we can use Theorem 1.1 for T to obtain the lowerbound L T > c / (cid:0) ns ( K ) /n (cid:1) . Combining this estimate with Theorem 1.2(ii), weimmediately get the reverse Santal´o inequality for symmetric bodies, and hence forall bodies. Theorem 1.3.
Let K be a symmetric convex body in R n . Then (1.10) s ( K ) /n > c n , where c > is an absolute constant. In Section 4 we briefly indicate how one can use Theorem 1.3 in order to es-tablish the existence of M -ellipsoids and the reverse Brunn–Minkowski inequality.The procedure is rather standard.The existence of an “ M -ellipsoid” associated with any centered convex body K in R n was proved by Milman in [19] (see also [20]): there exists an absoluteconstant c > K in R n we can find anorigin symmetric ellipsoid E K satisfying | K | = |E K | and1 c |E K + T | /n | K + T | /n c |E K + T | /n , (1.11) 1 c |E ◦ K + T | /n | K ◦ + T | /n c |E ◦ K + T | /n , for every convex body T in R n . The existence of M -ellipsoids can be equivalentlyestablished by introducing the M -position of a convex body. To any given centeredconvex body K in R n we can apply a linear transformation and find a position˜ K = u K ( K ) of volume | ˜ K | = | K | such that (1.11) is satisfied with E K a multipleof B n . This is the so-called M -position of K . It follows then that for every pair ofconvex bodies K and K in R n and for all t , t > | t ˜ K + t ˜ K | /n c ′ (cid:16) t | ˜ K | /n + t | ˜ K | /n (cid:17) , here c ′ > K or ˜ K (or both) by their polars. This statement is Milman’s reverse Brunn-Minkowski inequality.Another way to define the M -position of a convex body is through coveringnumbers. Recall that the covering number N ( A, B ) of a body A by a second body B is the least integer N for which there exist N translates of B whose union covers A . Then, as Milman proved, there exists an absolute constant β > K in R n has a linear image ˜ K which satisfies | ˜ K | = | B n | and(1.13) max { N ( ˜ K, B n ) , N ( B n , ˜ K ) , N ( ˜ K ◦ , B n ) , N ( B n , ˜ K ◦ ) } exp( βn ) . We say that a convex body K which satisfies (1.13) is in M -position with constant β . If K and K are two such convex bodies, there is a standard way to show thatthey and their polar bodies satisfy the reverse Brunn–Minkowski inequality (1.12)(see the end of Section 4). Note that M -ellipsoids and the M -position of a convexbody are not uniquely defined; see [2] for a recent description in terms of isotropicrestricted Gaussian measures.Pisier (see [26] and [27, Chapter 7]) has proposed a different approach to theseresults, which allows one to find a whole family of special M -ellipsoids satisfyingstronger entropy estimates. The precise statement is as follows. For every 0 < α < K in R n , there exists a linear image ˜ K of K whichsatisfies | ˜ K | = | B n | and(1.14) max { N ( ˜ K, tB n ) , N ( B n , t ˜ K ) , N ( ˜ K ◦ , tB n ) , N ( B n , t ˜ K ◦ ) } exp (cid:18) c ( α ) nt α (cid:19) for every t >
1, where c ( α ) is a constant depending only on α , with c ( α ) = O (cid:0) (2 − α ) − (cid:1) as α →
2. We then say that ˜ K is in M -position of order α (or α -regular M -position). It is an interesting question to give an elementary proof of the existenceof, say, an 1-regular M -position. Another interesting question is to check if theisotropic position is α -regular for some α > L K ≃ . As mentioned at the beginning of the Introduction, wedenote the Euclidean norm on R n by k · k . More generally, if K is a convex bodyin R n which contains 0 in its interior, then we write p K for its Minkowski functionalwhich is defined as follows:(2.1) p K ( x ) := inf { r > x ∈ rK } , x ∈ R n . If K is symmetric, we also write k · k K instead of p K . For every q > B , we define(2.2) I q ( K, B ) := (cid:18) | K | qn Z K k x k qB dx (cid:19) /q . f B is the Euclidean ball B n and K is an isotropic convex body in R n , then from(1.6) we see that(2.3) I ( K, B n ) = Z K k x k dx = Z K (cid:16) n X i =1 h x, e i i (cid:17) dx = nL K , so L K = I ( K, B n ) / √ n . More generally, as was explained in the Introduction, if K is an arbitrary convex body in R n , and we write ˜ K for the translate of K which iscentered, ˜ K = K − bar( K ), then the isotropic constant L K of K can be defined by(2.4) L K := 1 √ n inf (cid:8) I (cid:0) T ( ˜ K ) , B n (cid:1) : T is an invertible linear transformation (cid:9) . In the sequel, we write B for the homothetic image of volume 1 of a convex body B ⊂ R n , i.e. B := B | B | /n .As a generalization to convex bodies, we also consider logarithmically concave(or log-concave) measures on R n . This more general approach is justified by a well-known and very fruitful idea of K. Ball from [1] which allows one to transfer resultsfrom the setting of convex bodies to the broader setting of log-concave measuresand vice versa. We write P [ n ] for the class of all Borel probability measures on R n which are absolutely continuous with respect to the Lebesgue measure. The densityof µ ∈ P [ n ] is denoted by f µ . A probability measure µ ∈ P [ n ] is called symmetric if f µ is an even function on R n . We say that µ ∈ P [ n ] is centered if for all θ ∈ S n − ,(2.5) Z R n h x, θ i dµ ( x ) = Z R n h x, θ i f µ ( x ) dx = 0 . A measure µ on R n is called log-concave if for any Borel subsets A and B of R n and any λ ∈ (0 , µ ( λA + (1 − λ ) B ) > µ ( A ) λ µ ( B ) − λ . A function f : R n → [0 , ∞ )is called log-concave if log f is concave on its support { f > } . It is known thatif a probability measure µ is log-concave and µ ( H ) < H ,then µ ∈ P [ n ] and its density f µ is log-concave (see [3]). Note that if K is a convexbody in R n , then the Brunn-Minkowski inequality implies that K is the densityof a log-concave measure.There is also a way to generalize the notion of the isotropic constant of a convexbody in the setting of log-concave measures. Set(2.6) k µ k ∞ = sup x ∈ R n f µ ( x ) . The isotropic constant of µ is defined by(2.7) L µ := (cid:18) k µ k ∞ R R n f µ ( x ) dx (cid:19) n [det Cov( µ )] n , where Cov( µ ) is the covariance matrix of µ with entries(2.8) Cov( µ ) ij := R R n x i x j f µ ( x ) dx R R n f µ ( x ) dx − R R n x i f µ ( x ) dx R R n f µ ( x ) dx R R n x j f µ ( x ) dx R R n f µ ( x ) dx in the case that µ is a centered probability measure, we can write more simplyCov( µ ) ij := R R n x i x j f µ ( x ) dx ). It is straightforward to see that this definitioncoincides with the original definition of the isotropic constant when f µ is the char-acteristic function of a convex body. In addition, any bounds that we have forthe isotropic constants of convex bodies continue to hold essentially in this moregeneral setting. This can be seen through the following construction: let µ ∈ P [ n ] and assume that 0 ∈ supp( µ ). For every p >
0, we define a set K p ( µ ) as follows:(2.9) K p ( µ ) := (cid:26) x ∈ R n : p Z ∞ f µ ( rx ) r p − dr > f µ (0) (cid:27) . The sets K p ( µ ) were introduced in [1] and allow us to study log-concave measuresusing convex bodies. K. Ball proved that if µ is log-concave, then K p ( µ ) is a convexbody. Moreover, if µ is centered, then K n +1 ( µ ) is also centered, and we can provethat(2.10) c L K n +1 ( µ ) L µ c L K n +1 ( µ ) for some constants c , c > n .For basic facts from the Brunn-Minkowski theory and the asymptotic theory offinite dimensional normed spaces, we refer to the books [31], [24] and [27].The letters c, c ′ , c , c etc. denote absolute positive constants whose value maychange from line to line. Whenever we write a ≃ b for two quantities a, b associatedwith convex bodies or measures on R n , we mean that we can find positive constants c , c , independent of the dimension n , such that c a b c a . Also, if K, L ⊆ R n ,we will write K ≃ L if there exist absolute positive constants c , c such that c K ⊆ L ⊆ c K .In the rest of the section, we collect several tools and results from the asymptotictheory of convex bodies which will be used in Section 3. . Let K, B be convex bodies in R n with B symmetric. We will give an estimate for the covering numbers N ( K, tB ), t >
0, in terms of the quantity(2.11) I ( K, B ) = 1 | K | n Z K k x k B dx. Lemma 2.1.
Let K be a convex body of volume in R n containing as an interiorpoint. For any symmetric convex body B in R n and any t > , one has (2.12) log N ( K, tB ) c nI ( K, B ) t + log 2 , where c > is an absolute constant. roof. We define a Borel probability measure on R n by(2.13) µ ( A ) = 1 c K Z A e − p K ( x ) dx, where p K is the Minkowski functional of K and c K = R R n exp( − p K ( x )) dx . A simplecomputation, based on the fact that { x ∈ R n : p K ( x ) t } = tK for any t > c K = n !.Let { x , . . . , x N } be a subset of K which is maximal with respect to the condi-tion k x i − x j k B > t for i = j . Then K ⊆ S i N ( x i + tB ), and hence N ( K, tB ) N .Let a >
0. Note that if we set y i = (2 a/t ) x i , by the subadditivity of p K and thefact that p K ( x i )
1, we have(2.14) µ ( y i + aB ) > c K Z aB e − p K ( x ) e − p K ( y i ) dx > e − a/t µ ( aB ) . The bodies y i + aB have disjoint interiors, therefore N e − a/t µ ( aB )
1. It followsthat(2.15) N ( K, tB ) e a/t ( µ ( aB )) − . Now, we choose a > µ ( aB ) > /
2. A simple computation shows that(2.16) J := Z R n k x k K dµ ( x ) = ( n + 1) I ( K, B ) . By Markov’s inequality, µ (2 JB ) > /
2, so if we choose a = 2 J , we get(2.17) N ( K, tB ) J/t ) (cid:0) n + 1) I ( K, B ) /t (cid:1) for every t > ✷ Remark 2.2. (i) In the case that B is the Euclidean ball B n and K is an isotropicconvex body, we have that I ( K, B ) √ nL K and therefore(2.18) log N ( K, tB n ) c ′ n / L K t for any t > t the estimate is trivially true, since every isotropicbody K satisfies the inclusion K ⊆ cnL K B n for some absolute constant c ). Given(1.7), this is essentially the best way we can apply Lemma 2.1 when B = B n . Thisversion of the lemma appeared in the Ph.D. Thesis of Hartzoulaki [11]. The ideaof using I ( K, B n ) as a parameter in entropy estimates for isotropic convex bodiescomes from [22]. It was also used in [17] for a proof of the low M ∗ -estimate in thecase of quasi-convex bodies.(ii) Knowing that we have for any set S ,(2.19) N ( S − S, B n ) = N ( S − S, B n − B n ) N ( S, B n ) , e can use (2.18) to also get an upper bound for the covering numbers of thedifference body of an isotropic convex body K by the Euclidean ball:(2.20) log N ( K − K, tB n ) c ′ n / L K t . (iii) Lemma 2.1 is also related to the problem of estimating the mean width of anisotropic convex body K , namely the parameter w ( K ) := R S n − h K ( θ ) dσ ( θ ) where h K is the support function of K and σ is the uniform probability measure on S n − .The best upper bound we have is w ( K ) cn / L K (there are several argumentsleading to this estimate; see [9] and the references therein). It is known (see e.g.[8, Theorem 5.6]) that an improvement of the form(2.21) log N ( K, tB n ) c ′ n / L K t δ (for some δ >
0) in (2.18) would immediately imply a better bound for w ( K ) inthe isotropic case.The next lemma allows us to bound the dual covering numbers N ( B n , tK ◦ ). Lemma 2.3.
Let K be a convex body in R n which contains in its interior. Forevery t > we set A ( t ) := t log N ( K, tB n ) and B ( t ) := t log N ( B n , tK ◦ ) . Then,one has (2.22) sup t> B ( t )
16 sup t> A ( t ) . In particular, if K is isotropic ( or a translate of an isotropic convex body which stillcontains in its interior ) , then (2.23) log N ( B n , tK ◦ ) log N (cid:0) B n , t ( K − K ) ◦ (cid:1) c n / L K t , where c > is an absolute constant.Proof. We use a well-known idea from [32] (see also [16, Section 3.3]). For any t > t K ◦ ) ∩ (4 K ) ⊆ tB n . Passing to the polar bodies we see that(2.24) B n ⊆ conv (cid:18) t K ◦ , t K (cid:19) ⊆ t K ◦ + 2 t K. We write N ( B n , tK ◦ ) N (cid:18) t K ◦ + 2 t K, tK ◦ (cid:19) = N (cid:18) t K, t K ◦ (cid:19) (2.25) N (cid:18) t K, B n (cid:19) N (cid:18) B n , t K ◦ (cid:19) = N (cid:18) K, t B n (cid:19) N ( B n , tK ◦ ) . aking logarithms we get(2.26) B ( t ) A ( t/
8) + 12 B (2 t ) , for all t >
0. This implies that(2.27) B := sup t> B ( t ) A, and the result follows. ✷ The last covering lemma is from [20] and shows that the volume | conv (cid:0) K ∪ L (cid:1) | ofthe convex hull of two convex bodies K and L is essentially bounded by N ( L, K ) | K | ,provided that L ⊆ bK for some “reasonable” b > Lemma 2.4.
Let L be a convex body and let K be a symmetric convex body in R n .Assume that L ⊆ bK for some b > . Then (2.28) (cid:12)(cid:12) conv (cid:0) K ∪ L (cid:1)(cid:12)(cid:12) enb N ( L, K ) | K | . Proof.
By the definition of N ≡ N ( L, K ), there exist x , . . . , x N ∈ R n such that( x i + K ) ∩ L = ∅ for every i = 1 , . . . , N , and(2.29) L ⊆ N [ i =1 ( x i + K ) . From the symmetry of K and the fact that L ⊆ bK , it follows that, for every i = 1 , . . . , N ,(2.30) x i ∈ L + K ⊆ (1 + b ) K. Now, for every α, β ∈ [0 ,
1] with α + β = 1, we have that αL + βK ⊆ N [ i =1 ( αx i + αK ) + βK = N [ i =1 (cid:0) αx i + ( α + β ) K (cid:1) (2.31) = N [ i =1 ( αx i + K ) , and therefore(2.32) conv( L ∪ K ) = [ α (cid:0) αL + (1 − α ) K (cid:1) ⊆ N [ i =1 [ α ( αx i + K ) . We set T = 2 n and consider ⌈ bT ⌉ numbers α j equidistributed in [0 , j =1 , . . . , ⌈ bT ⌉ . From (2.30) and (2.32) it follows that: for every z ∈ conv( L ∪ K )there exist α, α j ∈ [0 , | α − α j | bT , such that(2.33) z ∈ αx i + K = α j x i + ( α − α j ) x i + K ⊆ α j x i + (cid:18) bbT + 1 (cid:19) K. e observe that(2.34) 1 + bbT = 1 + b nb n because b > b b , so (2.33) gives us that(2.35) z ∈ α j x i + (cid:18) n (cid:19) K. Going back to (2.32), we see that(2.36) conv( L ∪ K ) ⊆ N [ i =1 ⌈ bT ⌉ [ j =1 (cid:26) α j x i + (cid:18) n (cid:19) K (cid:27) . Then, | conv( L ∪ K ) | N ⌈ bT ⌉ (cid:18) n (cid:19) n | K | bT eN | K | (2.37) = 3 enb N ( L, K ) | K | , which is our claim. ✷ . In [12] Klartag gave an affirmativeanswer to the following question: even if we don’t know that every convex bodyin R n has bounded isotropic constant, given a body K can we find a second body T “geometrically close” to K with isotropic constant L T ≃
1? Here when we saythat K and T are “geometrically close”, we will mean that there exists an absoluteconstant c > x, y ∈ R n ,(2.38) 1 c ( T − x ) ⊆ K − y ⊆ c ( T − x ) . The method Klartag used is based on two key observations. The first one is that inorder to find a body T close to K which has bounded isotropic constant, it sufficesto define a positive log-concave function on K (vanishing everywhere else) withbounded isotropic constant and the extra property that its range is not too large. Proposition 2.5.
Let K be a convex body in R n and let f : K → (0 , ∞ ) be alog-concave function such that (2.39) sup x ∈ K f ( x ) m n inf x ∈ K f ( x ) for some m > . Let x be the barycenter of f , i.e. x = R R n xf ( x ) dx/ R R n f ( x ) dx ,and set g ( x ) = f ( x + x ) . Then, for the centered convex body T := K n +1 ( g ) , definedas in (2 . , we have that L f ≃ L T and (2.40) 1 m T ⊆ K − x ⊆ mT. he second observation is that a family of suitable candidates for the function f we need so as to apply Proposition 2.5 can be found through the logarithmicLaplace transform on K . In general, the logarithmic Laplace transform of a finiteBorel measure µ on R n is defined by(2.41) Λ µ ( ξ ) = log (cid:18)Z R n e h ξ,x i dµ ( x ) µ ( R n ) (cid:19) . In [12], Klartag makes use of the following properties of Λ µ : Proposition 2.6.
Let µ = µ K denote the Lebesgue measure on some convex body K in R n . Then, (2.42) (cid:0) ∇ Λ µ (cid:1) ( R n ) = int( K )( actually, for the arguments in [12] and for our proof here, it suffices to know that (cid:0) ∇ Λ µ (cid:1) ( R n ) ⊆ K ) . If µ ξ is the probability measure on R n with density proportionalto the function e h ξ,x i K ( x ) , then (2.43) b ( µ ξ ) = ∇ Λ µ ( ξ ) and Hess (Λ µ ( ξ )) = Cov( µ ξ ) . Moreover, the map ∇ Λ µ , which is one-to-one, transports the measure ν with density det Hess (Λ µ ) to µ . In other words, for every continuous non-negative function φ : R n → R , (2.44) Z K φ ( x ) dx = Z R n φ ( ∇ Λ µ ( ξ )) det Hess(Λ µ ( ξ )) dξ = Z R n φ ( ∇ Λ µ ( ξ )) dν ( ξ ) . Klartag’s approach has been recently applied in [6] where Dadush, Peikert andVempala provide an algorithm for enumerating lattice points in a convex body,with applications to integer programming and problems about lattice points. Theyuse the techniques of [12] in order to give an expected 2 O ( n ) -time algorithm forcomputing an M -ellipsoid for any convex body in R n . We now prove the reverse Santal´o inequality using the results that were describedin Section 2. The proof consists of three steps which roughly are the following:(i) we obtain a lower bound for the volume product s ( K ) which is optimal upto the value of the isotropic constant L K of K , (ii) by adapting Klartag’s mainargument from [12] we show that every symmetric convex body K has boundedgeometric distance (in the sense defined in (2.38)) from a second convex body T whose isotropic constant L T can be expressed in terms of s ( K ), and (iii) we usethe lower bound for s ( T ) in terms of L T , and the fact that s ( K ) and s ( T ) arecomparable, to get a lower bound for s ( K ) in which L K does not appear anymore. . Our first step will be toprove the following lower bound for s ( K ). roposition 3.1. Let K be a convex body in R n which contains in its interior.Then (3.1) 4 | K | /n | nK ◦ | /n > | K − K | /n | n ( K − K ) ◦ | /n > c L K , where c > is an absolute constant.Proof. We may assume that | K | = 1. From the Brunn-Minkowski inequality andthe classical Rogers–Shephard inequality (see [29]), we have 2 | K − K | /n K − K ) ◦ ⊆ K ◦ , we immediately see that(3.2) | K | /n | nK ◦ | /n > | K − K | /n | n ( K − K ) ◦ | /n , so it remains to prove the second inequality. Since(3.3) (cid:12)(cid:12) T ( K ) − T ( K ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) T ( K ) − T ( K ) (cid:1) ◦ (cid:12)(cid:12) = | K − K || ( K − K ) ◦ | for any invertible affine transformation T of K , we may assume for the rest of theproof that K is isotropic. We define(3.4) K := K − KL K ∩ B n and observe that the inclusion K ⊆ B n implies that B n ⊆ c nK ◦ for some absoluteconstant c . Moreover,(3.5) nK ◦ ≃ conv { nL K ( K − K ) ◦ , B n } , therefore we can apply Lemma 2.4 with L = B n and K = nL K ( K − K ) ◦ to bound | nK ◦ | from above; note that in this case b ≃ √ n , because K − K ⊆ cnL K B n since we have assumed K isotropic (see [7, Theorem 1.2.4]), and hence B n ⊆ c ′ √ n (cid:0) nL K ( K − K ) ◦ (cid:1) for some absolute constants c, c ′ . Using also (2.23) fromLemma 2.3 (with t ≃ √ nL K ), we see that c − n | nK ◦ | c n | conv { nL K ( K − K ) ◦ , B n }| (3.6) c n n / | nL K ( K − K ) ◦ | N (cid:16) B n , nL K ( K − K ) ◦ (cid:17) c n n / | nL K ( K − K ) ◦ | N (cid:0) B n , c √ nL K ( K − K ) ◦ (cid:1) e c n | nL K ( K − K ) ◦ | . This shows that there exists an absolute constant c ′ so that(3.7) | nL K ( K − K ) ◦ | /n > c ′ , and since | K − K | /n >
2, we have proven that(3.8) | K − K | /n | ( K − K ) ◦ | /n > c ′ nL K . ✷ . Our second step will be to show thatevery convex body K in R n has bounded geometric distance from a second convexbody T whose isotropic constant L T can be bounded in terms of s ( K − K ). roposition 3.2. Let K be a convex body in R n . For every ε ∈ (0 , there exista centered convex body T ⊂ R n and a point x ∈ R n such that (3.9) 11 + ε T ⊆ K + x ⊆ (1 + ε ) T and (3.10) L T c p εns ( K − K ) /n , where c > is an absolute constant.Proof. We may assume that K is centered and that | K − K | = 1. Indeed, oncewe prove the proposition for ˜ K := ( K − bar( K )) / | K − K | /n and some ε ∈ (0 , T which satisfies (3.9) and (3.10) with ˜ K instead of K , itwill immediately hold that the pair ( K, | K − K | /n T ) also satisfies these properties,because L T and s ( K − K ) are affine invariants.Recall from Proposition 2.6 that if µ = µ K is the Lebesgue measure restrictedon K , then the function ∇ Λ µ transports the measure ν with density(3.11) dνdξ = det Hess (Λ µ ( ξ )) ≡ det Cov( µ ξ )to µ . This implies that(3.12) ν ( R n ) = Z R n det Hess (Λ µ ( ξ )) dξ = Z K dx = | K | | K − K | = 1 . Thus, for every ε > | εn ( K − K ) ◦ | min ξ ∈ εn ( K − K ) ◦ det Cov( µ ξ ) Z εn ( K − K ) ◦ det Cov( µ ξ ) dξ = ν ( εn ( K − K ) ◦ ) , which means that there exists ξ ∈ εn ( K − K ) ◦ such that(3.14)det Cov( µ ξ ) = min ξ ′ ∈ εn ( K − K ) ◦ det Cov( µ ξ ′ ) | εn ( K − K ) ◦ | − = (cid:0) εns ( K − K ) /n (cid:1) − n (where the last equality holds because | K − K | = 1). Now, from the definition of µ ξ and (2.7) we have that(3.15) L µ ξ = (cid:18) sup x ∈ K e h ξ,x i R K e h ξ,x i dx (cid:19) n [det Cov( µ ξ )] n . Since ξ ∈ εn ( K − K ) ◦ and K ∪ ( − K ) ⊂ K − K , we know that |h ξ, x i| εn for all x ∈ K , therefore sup x ∈ K e h ξ,x i exp( εn ). On the other hand, since K is centered,from Jensen’s inequality we have that(3.16) 1 | K | Z K e h ξ,x i dx > exp (cid:18) | K | Z K h ξ, x i dx (cid:19) = 1 , hich means that R K e h ξ,x i dx > | K | > − n | K − K | by the Rogers-Shephard inequal-ity. Combining all these we get(3.17) L µ ξ e ε p εns ( K − K ) /n . Finally, we note that the function f ξ ( x ) = e h ξ,x i K ( x ) (which is proportional to thedensity of µ ξ ) is obviously log-concave and satisfies(3.18) sup x ∈ supp( f ξ ) f ξ ( x ) e εn inf x ∈ supp( f ξ ) f ξ ( x )(since |h ξ, x i| εn for all x ∈ K ). Therefore, applying Proposition 2.5, we can finda centered convex body T ξ in R n such that(3.19) L T ξ ≃ L f ξ = L µ ξ e ε p εns ( K − K ) /n and(3.20) 1 e ε T ξ ⊆ K − b ξ ⊆ e ε T ξ where b ξ is the barycenter of f ξ . Since e ε c ε when ε ∈ (0 , ✷ . Combining the previous two results wecan remove the isotropic constant L K from the lower bound for s ( K ) /n . Theorem 3.3.
Let K be a convex body in R n which contains in its interior.Then (3.21) | K | /n | nK ◦ | /n > c , where c > is an absolute constant.Proof. Since | K | /n | nK ◦ | /n > | K − K | /n | n ( K − K ) ◦ | /n , we may assume forthe rest of the proof that K is symmetric. Using Proposition 3.2 with ε = 1 /
2, wefind a convex body T ⊂ R n and a point x ∈ R n such that(3.22) 23 T ⊆ K + x ⊆ T and L T c / p ns ( K ) /n for some absolute constant c >
0. Proposition 3.1 showsthat(3.23) | T − T | /n | n ( T − T ) ◦ | /n > c L T , here c > ( T − T ) ⊆ K − K = 2 K ⊆ ( T − T ), and thus K ◦ ⊇ ( T − T ) ◦ . Therefore, combining the above, we get ns ( K ) /n = | nK ◦ | /n | K | /n > | n ( T − T ) ◦ | /n | T − T | /n (3.24) > c ′ L T > c q ns ( K ) /n , and so it follows that(3.25) s ( K ) /n > c n with c = c . This completes the proof. ✷ Having proved the reverse Santal´o inequality, one can go back to Proposition3.2 and insert the lower bound for s ( K − K ). This is the last step in Klartag’ssolution of the isomorphic slicing problem. Theorem 3.4 (Klartag) . Let K be a convex body in R n . For every ε ∈ (0 , thereexist a centered convex body T ⊂ R n and a point x ∈ R n such that (3.26) 11 + ε T ⊆ K + x ⊆ (1 + ε ) T and (3.27) L T c √ ε , where c > is an absolute constant. M -ellipsoids and the reverse Brunn-Minkowskiinequality We can now prove the existence of M -ellipsoids for any convex body and, as aconsequence, the reverse Brunn–Minkowski inequality. M -ellipsoids . Let K be a centered convex body in R n . Wewill give a proof of the existence of an M -ellipsoid for K . The next Proposition isthe first step. Proposition 4.1.
Let K be a centered convex body in R n . Then there exists anellipsoid E K such that | K | = |E K | and (4.1) max { log N ( K, t E K ) , log N ( E ◦ K , tK ◦ ) } cnt for all t > , where c > is an absolute constant. roof. Applying Proposition 3.4, we can find a centered convex body T withisotropic constant L T C such that(4.2) 23 T ⊆ K + x ⊆ T for some x ∈ R n . Let Q ( T ) be an isotropic position of T . From Remark 2.2(ii) andLemma 2.3 we know that(4.3) max { log N (cid:0) Q ( T ) − Q ( T ) , t √ nB n (cid:1) , log N (cid:0) B n , t √ n ( Q ( T ) − Q ( T )) ◦ (cid:1) } cnt for every t >
0. Since(4.4) 23 ( Q ( T ) − Q ( T )) ⊆ Q ( K ) − Q ( K ) ⊆
32 ( Q ( T ) − Q ( T ))and Q ( K ) ⊆ Q ( K ) − Q ( K ) , ( Q ( K ) − Q ( K )) ◦ ⊆ ( Q ( K )) ◦ , from (4.3) it follows that(4.5) max { log N (cid:0) Q ( K ) , t √ nB n (cid:1) , log N (cid:0) B n , t √ n ( Q ( K )) ◦ (cid:1) } c ′ nt for every t >
0. We define E K := Q − ( a √ nB n ) where a is chosen so that | Q ( K ) | = | a √ nB n | (equivalently, so that |E K | = | K | ), and from (4.3) we get that(4.6) max { log N ( K, t E K ) , log N ( E ◦ K , tK ◦ ) } c ′ ant for all t >
0. It remains to observe that(4.7) |√ nB n | /n ≃ | Q ( T ) | /n ≃ | Q ( K + x ) | /n = | Q ( K ) | /n , whence it follows that a ≃ ✷ We now recall some standard entropy estimates which are valid for arbitraryconvex bodies in R n . Lemma 4.2.
Let K and L be convex bodies in R n . If L is symmetric, then (4.8) N ( K, L ) | K + L/ || L/ | n | K + L || L | , whereas in the general case (4.9) N ( K, L ) n | K + L || L | . Moreover, (4.10) | K + L || L | n N ( K, L ) . roof. The proof of (4.10) is an easy consequence of the definitions. To prove (4.8),note that if N is a maximal subset of K with respect to the property(4.11) x, y ∈ N and x = y ⇒ k x − y k L > , then K ⊆ S x ∈ N ( x + L ), while every two sets x + L/ , y + L/ x, y ∈ N ) havedisjoint interiors when x = y .Finally, when L is not necessarily symmetric, we recall that N ( K + x, L + y ) = N ( K, L ) for every x, y ∈ R n , and also that the ratio | K + L | / | L | obviously remainsunaltered if we translate K or L . Hence, we can assume that L is centered, in whichcase it follows from [23, Corollary 3] that(4.12) | L ∩ ( − L ) | > − n | L | . But then, from (4.8) we get that(4.13) N ( K, L ) N ( K, L ∩ ( − L )) n | K + ( L ∩ ( − L )) || L ∩ ( − L ) | n | K + L || L | , and we have (4.9). ✷ Corollary 4.3.
Let K and L be two convex bodies in R n . Then, (4.14) N ( K, L ) /n ≃ | K + L | /n | L | /n . It also follows that if K and L have the same volume, then (4.15) N ( K, L ) /n N ( L, K ) /n . Combining Proposition 4.1 with the classical Santal´o inequality and Corollary4.3, we can now prove the existence of M -ellipsoids for any centered convex bodyin R n . Theorem 4.4.
Let K be a centered convex body in R n . There exists an ellipsoid E K such that | K | = |E K | and (4.16) max (cid:8) log N ( K, E K ) , log N ( E K , K ) , log N ( K ◦ , E ◦ K ) , log N ( E ◦ K , K ◦ ) (cid:9) cn, where c > is an absolute constant.Proof. Let E K be the ellipsoid defined in Proposition 4.1. It immediately followsthat(4.17) max (cid:8) N ( K, E K ) , N ( E ◦ K , K ◦ ) (cid:9) exp( cn ) . For the other two covering numbers we use Lemma 4.2: N ( E K , K ) n N ( K, E K ),which means that log N ( E K , K ) (log 8) n + log N ( K, E K ). Similarly,(4.18) N ( K ◦ , E ◦ K ) n | K ◦ + E ◦ K ||E ◦ K | n | K ◦ + E ◦ K || K ◦ | n N ( E ◦ K , K ◦ ) , here we have also used the fact that | K | = |E K | ⇒ | K ◦ | |E ◦ K | from the classicalSantal´o inequality. This completes the proof. ✷ . As a consequence of Theorem 4.4and Corollary 4.3, we get the “reverse” Brunn-Minkowski inequality. Theorem 4.5.
Let K be a centered convex body in R n . There exists an ellipsoid E K such that | K | = |E K | and for every convex body T in R n , e − ( c +log8) |E K + T | /n | K + T | /n e c +log8 |E K + T | /n , (4.19) e − ( c +log8) |E ◦ K + T | /n | K ◦ + T | /n e c +log8 |E ◦ K + T | /n , (4.20) where c is the constant we found in Theorem .Proof. Let E K be the ellipsoid defined in Proposition 4.1. Using Lemma 4.2, wecan write |E K + T | /n | T | /n N ( E K , T ) /n | T | /n N ( E K , K ) /n N ( K, T ) /n (4.21) e c | T | /n N ( K, T ) /n e c | K + T | /n . The same reasoning gives us the second part of (4.19) and (4.20). ✷ Remark 4.6.
We usually say that a centered convex body K is in M -position ifthe ellipsoid E K that we look for in Theorem 4.4 can be taken to be a multiple ofthe Euclidean ball. Obviously, if r K := | K | /n / | B n | /n and E K = T K ( r K B n ) forsome volume-preserving T K , then ˜ K := T − K ( K ) is a linear image of K of the samevolume which is in M -position. Assume then that ˜ K and ˜ K are two such imagesof some bodies K and K in R n , and that K ′ i stands for either ˜ K i or ( ˜ K i ) ◦ . Using(4.19) and (4.20), we see that | K ′ + K ′ | /n c | K ′ + r K ′ B n | /n c | r K ′ B n + r K ′ B n | /n (4.22) = c (cid:0) r K ′ + r K ′ (cid:1) | B n | /n = c (cid:0) | K ′ | /n + | K ′ | /n (cid:1) . This means that we have a partial inverse to the Brunn-Minkowski inequality whichholds true for certain affine images of any convex bodies K , K and the polars ofthose images. A direct consequence of (4.22) and Corollary 4.3 is the following: Corollary 4.7.
Let K and L be two convex bodies in R n of the same volume whichare in M -position. Then, (4.23) N ( K, tL ) /n ≃ N ( L, tK ) /n for every t > .Proof. Since tL and tK are also in M -position for every t >
0, we have that N ( K, tL ) /n ≃ | K + tL | n | tL | n ≃ | K | n + t | L | n t | L | n (4.24) = t | K | n + | L | n t | K | n ≃ | tK + L | n | tK | n ≃ N ( L, tK ) /n . ✷ inally, let us remark that, as Pisier notes in [27], the asymptotic form of theSantal´o inequality and its inverse and the existence of an M -position for any convexbody are interconnected results: if we know that for every centered convex body K there exists an ellipsoid E K such that(4.25) max (cid:8) log N ( K, E K ) , log N ( E K , K ) , log N ( K ◦ , E ◦ K ) , log N ( E ◦ K , K ◦ ) (cid:9) cn for some absolute constant c >
0, then we can prove that(4.26) e − c +log8) s ( B n ) s ( K ) e c +log8) s ( B n )for all centered bodies K . Indeed, if E K is an M -ellipsoid for K as above, thenfrom Lemma 4.2, |E K + K | /n | K | /n N ( E K , K ) /n e c e c N ( K, E K ) /n e c |E K + K | /n |E K | /n , so |E K | /n e c | K | /n , and in the same manner,(4.27) max n | K | /n |E K | /n , |E ◦ K | /n | K ◦ | /n , | K ◦ | /n |E ◦ K | /n o e c . (4.26) now follows. Acknowledgements . The second named author wishes to thank the A. SloanFoundation and the US National Science Foundation for support through the grantDMS-0906150. The third named author is supported by a scholarship of the Uni-versity of Athens.
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Apostolos Giannopoulos : Department of Mathematics, National and KapodistrianUniversity of Athens, Panepistimioupolis 157-84, Athens, Greece.
E-mail: [email protected]
Grigoris Paouris : Department of Mathematics, Texas A & M University, College Sta-tion, TX 77843 U.S.A.
E-mail: [email protected]
Beatrice-Helen Vritsiou : Department of Mathematics, National and KapodistrianUniversity of Athens, Panepistimioupolis 157-84, Athens, Greece.
E-mail: [email protected]@math.uoa.gr