We develop a reverse entropy power inequality for convex measures, which may be seen as an affinegeometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this… (More)

The classical entropy power inequality is extended to the Rényi entropy. We also discuss the question of the existence of the entropy for sums of independent random variables.

A generalization of Young’s inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter… (More)

The entropy per coordinate in a log-concave random vector of any dimension with given density at the mode is shown to have a range of just 1. Uniform distributions on convex bodies are at the lower… (More)

We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian… (More)

Here #n is the standard Gaussian measure in R, of density d#n(x)=>k=1 ,(xk) dxk , x=(x1 , . . ., xn) # R , ,(xk)=1 2? exp(&xk 2), 8 is the inverse of the distribution function 8 of #1 , and A=[x # R:… (More)

In an important paper, Alon [2] derived a Cheeger–type inequality [8], by bounding from below the second smallest eigenvalue of the Laplacian of a finite undirected graph by a function of a (vertex)… (More)

The hyperplane conjecture is a major unsolved problem in high-dimensional convex geometry that has attracted much attention in the geometric and functional analysis literature. It asserts that there… (More)

We present a simple direct proof of the classical Sobolev inequality in Rn with best constant from the geometric Brunn–Minkowski–Lusternik inequality. Mathematics Subject Classification 46-XX