*Published Paper*

**Inserted:** 19 jun 2012

**Last Updated:** 17 dec 2013

**Year:** 2013

**Doi:** 10.1007/s10231-013-0334-x

**Abstract:**

Let $K \subset \mathbb R^N$ be a convex body containing the origin. A measurable set $G \subset \mathbb R^N$ 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 $C^{1,1}$-regular; also, if $K$ is centrally symmetric, $K$ must be strictly convex, $C^{1,1}$-regular and such that $K = G − G$ up to homotheties; this implies in turn that $G$ must be $C^{2,1}$- 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).

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