Characterization of ellipsoids as K-dense sets

Let $K\subset\RN$ be any convex body containing the origin. A measurable set $G\subset\RN$ 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 \cite{MM}, 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 $\RN$: 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 \cite{MM}, 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 \cite{Pe}.