In the paper we study closures of classes of log–concave measures under taking weak limits, linear transformations and tensor products. We consider what uniform measures on convex bodies can one obtain starting from some class K. In particular we prove that if one starts from one–dimensional log–concave measures, one obtains no non– trivial uniform mesures on convex bodies.

We prove new entropy inequalities for log concave and s-concave functions that strengthen and generalize recently established reverse log Sobolev and Poincaré inequalities for such functions. This leads naturally to the concept of f -divergence and, in particular, relative entropy for s-concave and log concave functions. We establish their basic properties, among them the affine invariant valua...

In contemporary convex geometry, the rapidly developing Lp-Brunn Minkowski theory is a modern analogue of the classical Brunn Minkowski theory. A cornerstone of this theory is the Lp-affine surface area for convex bodies. Here, we introduce a functional form of this concept, for log concave and s-concave functions. We show that the new functional form is a generalization of the original Lp-affi...

We study the expected volume of random polytopes generated by taking convex hull independent identically distributed points from a given distribution. show that for log-concave distributions supported on bodies, we need at least exponentially many (in dimension) samples to be significant and super-exponentially suffice concave measures when their parameter concavity is positive.

Abstract John’s fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ $\mathbb{R}^{d}$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids positions (affine images) of another $L$. Another, more recent direction consider logarithmically concave functions on instead bodies: we designate some spec...

Determining the minimum distance between two convex objects is a problem that has been solved using many di¤erent approaches. Some methods rely on computational geometry techniques, while others rely on optimization techniques to ...nd the solution. Some fast algorithms sacri...ce precision for speed while others are limited in the types of objects that they can handle (e.g. linearly bound obje...

We introduce Reflective Hamiltonian Monte Carlo (ReHMC), an HMC-based algorithm, to sample from a log-concave distribution restricted convex body. prove that, starting warm start, the walk mixes target $\pi(x) \propto e^{-f(x)}$, where $f$ is $L$-smooth and $m$-strongly-convex, within accuracy $\varepsilon$ after $\widetilde O(\kappa d^2 \ell^2 \log (1 / \varepsilon))$ steps for well-rounded bo...