A remark on A+B and A-A for compact sets in R^n

TL;DR

This paper establishes a new measure inequality relating the difference set A-A and the sum set A+B for compact sets in R^n, highlighting the influence of set measures and dimension.

Contribution

It provides a novel measure inequality connecting A-A and A+B for compact sets in R^n, extending understanding of set operations in geometric measure theory.

Findings

The measure of A-A is bounded by the square of the measure of A+B divided by the product of sqrt(n) and the measure of A.

The inequality holds when B is a compact set with measure at least that of A.

The result applies to convex sets A and arbitrary compact sets B in R^n.

Abstract

We prove in particular that if A be a compact convex subset of R^n, and B from R^n be an arbitrary compact set then \mu (A-A) \ll \mu(A+B)^2 / (\sqrt{n} \mu (A)), provided that \mu(B)\ge \mu(A).