Fractional generalizations of Young and Brunn-Minkowski inequalities

TL;DR

This paper proposes a fractional generalization of Young and Brunn-Minkowski inequalities, providing proofs for certain cases and connecting to entropy power inequalities and the law of large numbers for random sets.

Contribution

It introduces a new fractional framework for Young and Brunn-Minkowski inequalities, proving the conjecture for convex sets and linking to entropy and random set laws.

Findings

Conjecture proven for certain parameter ranges.

Generalized Brunn-Minkowski conjecture holds for convex sets.

Application to the law of large numbers for random sets.

Abstract

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 ranges. The conjecture would provide a unified proof of recent entropy power inequalities of Barron and Madiman, as well as of a (conjectured) generalization of the Brunn-Minkowski inequality. It is shown that the generalized Brunn-Minkowski conjecture is true for convex sets; an application of this to the law of large numbers for random sets is described.