Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke-Santalo inequality and the affine isoperimetric inequalities for many sets and many functions. We derive from it an ent
ropy bound for the total Kantorovich cost appearing in the barycenter problem. We also establish a pointwise Prekopa-Leindler inequality and show a monotonicity property of the multimarginal Blaschke-Santalo functional.
In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional affine su
rface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities lead to new inequalities for L_p-affine surface areas for convex bodies.
Let $gamma$ be the standard Gaussian measure on $mathbb{R}^n$ and let $mathcal{P}_{gamma}$ be the space of probability measures that are absolutely continuous with respect to $gamma$. We study lower bounds for the functional $mathcal{F}_{gamma}(mu) =
{rm Ent}(mu) - frac{1}{2} W^2_2(mu, u)$, where $mu in mathcal{P}_{gamma}, u in mathcal{P}_{gamma}$, ${rm Ent}(mu) = int logbigl( frac{mu}{gamma}bigr) d mu$ is the relative Gaussian entropy, and $W_2$ is the quadratic Kantorovich distance. The minimizers of $mathcal{F}_{gamma}$ are solutions to a dimension-free Gaussian analog of the (real) Kahler-Einstein equation. We show that $mathcal{F}_{gamma}(mu) $ is bounded from below under the assumption that the Gaussian Fisher information of $ u$ is finite and prove a priori estimates for the minimizers. Our approach relies on certain stability estimates for the Gaussian log-Sobolev and Talagrand transportation inequalities.
Given the standard Gaussian measure $gamma$ on the countable product of lines $mathbb{R}^{infty}$ and a probability measure $g cdot gamma$ absolutely continuous with respect to $gamma$, we consider the optimal transportation $T(x) = x + abla varphi(
x)$ of $g cdot gamma$ to $gamma$. Assume that the function $| abla g|^2/g$ is $gamma$-integrable. We prove that the function $varphi$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g = {det}_2(I + D^2 varphi) exp bigl(mathcal{L} varphi - 1/2 | abla varphi|^2 bigr)$. We also establish sufficient conditions for the existence of third order derivatives of $varphi$.
Let $A subset mathbb{R}^d$, $dge 2$, be a compact convex set and let $mu = varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $ u = varrho_1 dx$ be a probability measure on $B_r := {xcolon |x| le r}$ e
quivalent to the restriction of Lebesgue measure. We prove that there exists a mapping $T$ such that $ u = mu circ T^{-1}$ and $T = phi cdot {rm n}$, where $phicolon A to [0,r]$ is a continuous potential with convex sub-level sets and ${rm n}$ is the Gauss map of the corresponding level sets of $phi$. Moreover, $T$ is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth $phi$ the level sets of $phi$ are driven by the Gauss curvature flow $dot{x}(s) = -s^{d-1} frac{varrho_1(s {rm n})}{varrho_0(x)} K(x) cdot {rm n}(x)$, where $K$ is the Gauss curvature. As a by-product one can reprove the existence of weak solutions of the classical Gauss curvature flow starting from a convex hypersurface.
