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

Abstract

Let (Formula presented.) be a convex body containing the origin. A measurable set (Formula presented.) with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r>0, the measure of (Formula presented.) is constant when x varies on the boundary of G (here, (Formula presented.) denotes a translation of a dilation of K). We first prove that G must always be strictly convex and at least (Formula presented.)-regular; also, if K is centrally symmetric, K must be strictly convex, (Formula presented.)-regular and such that (Formula presented.) up to homotheties; this implies in turn that G must be (Formula presented.)-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 et al. in 2008. 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 (Formula presented.) for the measure of (Formula presented.) (needed in 2008).