Characterization of ellipsoids as K-dense sets

Let K\subset R^N be any convex body containing the origin. A measurable set G\subset R^N with finite and positive Lebesgue measure is said to be K-dense if, for any fixed r>0, the measure of G\cap (x+r K) is constant when x varies on the boundary of G (here, x+r K denotes a translation of a dilation of K). In [6], we proved for the case in which N=2 that if G is K-dense, then both G and K must be homothetic to the same ellipse. Here, we completely characterize K-dense sets in R^N: if G is K-dense, then both G and K must be homothetic to the same ellipsoid. Our proof, by building upon results obtained in [6], relies on an asymptotic formula for the measure of G\cap (x+r K) for large values of the parameter r and a classical characterization of ellipsoids due to C.M. Petty [9].