  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].
