Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

TL;DR

This paper characterizes ellipses as the only uniformly dense sets with respect to convex bodies, proving regularity and symmetry properties, and extending previous results to higher dimensions without regularity assumptions.

Contribution

It establishes new regularity conditions for uniformly K-dense sets and generalizes the characterization of ellipses to higher dimensions without regularity constraints.

Findings

G must be strictly convex and C1,1-regular.

If K is centrally symmetric, then K and G are homothetic to the same ellipse.

The characterization extends to higher dimensions using Minkowski's and affine inequalities.

Abstract

Let K \subset R^N be a convex body containing the origin. A measurable set G \subset R^N with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r > 0, the measure of G \cap (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). We first prove that G must always be strictly convex and at least C1,1-regular; also, if K is centrally symmetric, K must be strictly convex, C1,1-regular and such that K = G - G up to homotheties; this implies in turn that G must be C2,1- regular. Then for N = 2, we prove that G is uniformly K-dense if and only if K and G are homothetic to the same ellipse. This result was already proven by Amar, Berrone and Gianni in [3]. However, our proof removes their regularity assumptions on K and G and, more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski's inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near r = 0 for the measure of G\cap(x+rK) (needed in [3]).