Characterization of Ellipses as Uniformly Dense Sets with Respect to a Family of Convex Bodies

Let K ⊂ R be a convex body containing the origin. A measurable set G ⊂ R with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r > 0, the measure of G ∩ (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-regular; also, if K is centrally symmetric, K must be strictly convex, C-regular and such that K = G − G up to homotheties; this implies in turn that G must be Cregular. 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 ∩ (x+ rK) (needed in [3]).