for any compact sets K, T ⊂ R, where (K +T )/2 = {(x+ y)/2; x ∈ K, y ∈ T} is half of the Minkowski sum of K and T , and where V oln stands for the Lebesgue measure in R. Equality in (1) holds if and only if K is a translate of T and both are convex, up to a set of measure zero. The literature contains various stability estimates for the Brunn-Minkowski inequality, which imply that when there is...

For 0 < p < 1, Haberl and Ludwig defined the notions of symmetric and asymmetric Lp-intersection bodies. Recently, Wang and Li introduced the general Lp-intersection bodies. In this paper, we give the Lp-dual geominimal surface area forms for the extremum values and Brunn-Minkowski type inequality of general Lp-intersection bodies. Further, combining with the Lp-dual geominimal surface areas, w...

A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinal...

Correspondence On the Similarity of the Entropy Power Inequality The preceeding equations allow the entropy power inequality and the Brunn-Minkowski Inequality to be rewritten in the equivalent form (4) where X' and Y' are independent normal variables with corresponding entropies H(X') = H(X) and H(Y') = H(Y). Verification of this restatement follows from the use of (1) to show that Abstract-Th...

The entropy power inequality, which plays a fundamental role in information theory and probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by this connection to Convex Geometry, we survey various recent developments on forward and reverse entropy power inequalities not just for the Shannon-Boltzmann entropy but also more generally for Rényi entropy. In the proce...

Corresponding to each origin-symmetric convex (or more general) subset of Euclidean n-space R, there is a unique ellipsoid with the following property: The moment of inertia of the ellipsoid and the moment of inertia of the convex set are the same about every 1-dimensional subspace ofR. This ellipsoid is called the Legendre ellipsoid of the convex set. The Legendre ellipsoid and its polar (the ...