The Logarithmic Laplace Transform in Convex Geometry Bo’az Klartag Tel-Aviv University Workshop at Kibbutz Hagoshrim, Israel, April 2011 Joint works with Ronen Eldan and with Emanuel Milman. Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry Many open problems We would like to explain some of the tools used in the study of the distribution of volume in high dimensional convex bodies. Despite recent progress, even the simplest questions remain unsolved: Question [Bourgain, 1980s] Suppose K ⊂ Rn is a convex body of volume one. Does there exist an (n − 1)-dimensional hyperplane H ⊂ Rn such that Voln−1(K \ H) > c where c > 0 is a universal constant? −1=4 Known: Voln−1(K \ H) > cn (Bourgain ’91, K. ’06). Affirmative answer for: unconditional convex bodies, zonoids, their duals, random convex bodies, outer finite volume ratio, few vertices/facets, subspaces/quotients of Lp, Schatten class, ::: Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry Logarithmically-Concave densities As was observed by K. Ball, the hyperplane conjecture is most naturally formulated in the class of log-concave densities. A probability density on Rn is log-concave if it takes the form exp(−H) for a convex function H : Rn ! (−∞; 1]. Examples of log-concave densities: The Gaussian density, the uniform density on a convex body. 1 Pointwise product of log-concave densities is (proportional to) a log-concave density. 2 Prékopa-Leindler: If X is a log-concave random vector, so is the random vector T (X) for any linear map T . Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry Isotropic Constant For a log-concave probability density ρ : Rn ! [0; 1) set 1 1 Lρ = sup ρ n (x) det Cov(ρ) 2n x2Rn the isotropic constant of ρ. The isotropic constant is affinely invariant. What’s its meaning? Normalization: Suppose X is a random vector in Rn with density ρ. It is isotropic if EX = 0; Cov(X) = Id That is, all marginals have mean zero and var. one. For an isotropic, log-concave density ρ in Rn, Z Z 1=n 1+ 1 1 Lρ ' ρ(0) ' ρ n ' exp ρ log ρ > c Rn n Rn Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry An equivalent formulation of the slicing problem The hyperplane conjecture is directly equivalent to the following: Slicing problem, again: Is it true that for any n and an isotropic, log-concave ρ : Rn ! [0; 1), Lρ < C where C > 0 is a universal constant? (this simple equivalence follows from Ball, Bourgain, Fradelizi, Hensley, Milman, Pajor and others, using Brunn-Minkowski). For a uniform density on K ⊂ Rn, −1=n LK = Voln(K ) . Can we have the same covariance as the Euclidean ball, in a substantially smaller convex set? Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry a Remarks 1 It is straightforward to show that Lρ > c, for a universal constant c > 0. 2 To summarize, define Ln = sup Lρ: ρ:Rn![0;1) It is currently known that 1=4 Ln ≤ Cn : 3 It is enough to consider the uniform measure on centrally-symmetric convex bodies (Ball ’88, K. ’05): Ln ≤ C sup LK K ⊂Rn where K ⊂ Rn is convex, with K = −K . Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry Our tool: Logarithmic Laplace Transform The Laplace Transform is used in almost all known bounds for Ln. Suppose X is a log-concave random vector, EX = 0. The logarithmic Laplace transform is the convex function n Λ(ξ) = ΛX (ξ) = log E exp(X · ξ)(ξ 2 R ): It is non-negative with Λ(0) = 0. The logarithmic Laplace transform helps relate: 1 Covariance matrix of X (and its “tilts”), and 2 The entropy of X (and volumes of Zp(X)). Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry Differentiating the logarithmic Laplace transform Recall that Λ(ξ) = log E exp(X · ξ) n ~ For ξ 2 R , denote by Xξ the “tilted” log-concave random vector in Rn whose density is proportional to x 7! ρ(x) exp(ξ · x) where ρ is the density of X. Then, ~ 1 rΛ(ξ) = EXξ. 2 ~ 2 The hessian r Λ(ξ) = Cov(Xξ). ~ Third derivatives? A bit complicated. With bξ = EXξ, @i log det r2Λ(ξ) h −1 ~ i i ~ ~ i = Tr Cov(Xξ) E(Xξ − bξ)(Xξ − bξ) ⊗ (Xξ − bξ) : Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry To tilt, or not to tilt? We prefer the “centered tilt”, d ~ Xξ = Xξ − bξ ~ where bξ = EXξ. Note that EXξ = 0, and Xξ is log-concave. Focusing on the Log. Laplace transform near ξ 2 Rn Note that ΛXξ (z) = ΛX (ξ + z) − [ΛX (ξ) + z · rΛX (ξ)] : i.e., we subtract the tangent plane at ξ, and translate so that ξ is the new origin. A marvelous usage of tilts appears in Cramér’s theorem on moderate deviations from 1938. Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry Transportation of measure Suppose X is uniform in a convex body K . Recall that rΛ(ξ) = bξ 2 K for all ξ, by convexity. From the change of variables formula, Z Z n 2 2 Voln(K ) = Voln(rΛ(R )) = det r Λ(ξ)dξ ≥ det r Λ Rn nK ◦ In particular, there exists ξ 2 nK ◦ with 2 Voln(K ) det r Λ(ξ) = det Cov(Xξ) ≤ ◦ : Voln(nK ) Since e−n ≤ exp(ξ · x) ≤ en for x 2 K , then for such ξ 2 nK ◦, C Vol (K ) 1=(2n) 1 1=(2n) L ≤ n ' : Xξ 1=n ◦ ◦ Voln(K ) Voln(nK ) Voln(K )Voln(nK ) Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry Log-concave densities and convex bodies Theorem (Bourgain-Milman ’87) ◦ n Voln(K )Voln(nK ) ≥ c where c > 0 is a universal constant. ◦ Therefore LXξ < Const for most ξ 2 nK . There is a correspondence between centered log-concave densities and convex bodies due to K. Ball: Suppose f : Rn ! [0; 1) is log-concave. Denote Z 1 n n K (f ) = x 2 R ;(n + 1) f (rx)r dr ≥ 1 ; 0 the convex body associated with f . (Convexity of K (f ) is related to Busemann’s inequality) Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry Isomorphic version of the slicing problem Applying this construction to Xξ with LXξ < C: Obtain a convex body T , with LT ' LXξ < Const. Direct analysis: The convex body T is geometrically close to K , the support of X. We deduce: Theorem (K. ’06) For any convex body K ⊂ Rn and 0 < " < 1, there exists another convex body T ⊂ Rn with 1 (1 − ")K ⊆ T ⊆ (1 + ")K, p 2 LT ≤ C= ", where C > 0 is a universal constant. Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry The Calculus of the Logarithmic Laplace Transform There are a few other useful tricks with Log. Laplace. Many properties of Log. Laplace are proven via 1D arguments 1 For a subspace E (perhaps of dimension one), use the fact that ΛX jE = ΛProjE (X), where ProjE (X) is again l.c. 2 Compute ΛX via integration in polar coordinates, and use R 1 n−1 Laplace method for the integrals 0 t (l:c:)dt. Suppose X is log-concave, EX = 0; t > 0. Denote n fΛ ≤ tg = fξ 2 R ; Λ(ξ) ≤ tg and \ fΛ ≤ tgsymm = fΛ ≤ tg (−{Λ ≤ tg) : Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry Direct Properties: (proven via 1D considerations) Using these methods, one shows: Suppose X is uniform in K and EX = 0. Then, fΛ ≤ ng ' nK ◦: For a general log-concave X with EX = 0, 1=(2n) 1=n LX ' (det Cov(X)) · Voln (fΛ ≤ ng) : A rather technical, but useful fact, which follows from LX ≥ c: For any k-dim. subspace E ⊂ Rn, there exists θ 2 E with jθj = 1 and q 2 −1=k E(X · θ) & Volk (fΛ ≤ kg \ E) : Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry Relation to Lp-centroid bodies of Paouris Suppose X is log-concave, EX = 0. For p ≥ 1 consider the norm p 1=p n hZp(X)(θ) = (EjX · θj ) (θ 2 R ): This is the supporting functional of a centrally-symmetric convex body, denoted by Zp(X) (introduced by Lutwak, Zhang ’97, theory developed mostly by Paouris). From 1D considerations: Theorem (essentially from Latała, Wojtaszczyk ’08) For any p ≥ 1, ◦ Zp(X) ' pfΛX ≤ pgsymm: Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry Borrowing from Paouris’ theory of Lp-centroid bodies One advantage of Zp(X) over the dual fΛX ≤ pg: Z 1=p p p p p 1=p p Wp(Zp(X)) := hZp (θ) ' p (EjXj ) & p Sn−1 n + p when 2 ≤ p ≤ n and EjXj2 = n. This is very hard to prove directly for log. Laplace. Paouris q∗(X) parameter This is the maximal p ≥ 1 such that random p-dimensional sections of fΛX ≤ pgsymm are approximately Euclidean. It is important to have the same p, because we know something about the geometry of fΛ ≤ ng in Rn. Bo’az Klartag The Logarithmic Laplace Transform in Convex Geometry The classical Dvoretzky theorem appears V. Milman’s 1971 proof of Dvoretzky’s theorem has a formula for the dimension in which “random sections are Euclidean”. Paouris remarkable theorem Using this formula and its relations to Wp(Zp(X)), he shows: 1=2 p 1=p 2 (EjXj ) ≤ C EjXj ∗ ∗ p for all 1 ≤ p ≤ q (X) and an isotropic X, with q (X) & n.