The volume of the intersection of a convex body with its translates

Mathieu Meyer, Shlomo Reisner, M. Schmuckenschlager

It is proved that for a symmetric convex body K in R^n, if for some tau > 0, |K cap (x+tau K)| depends on ||x||_K only, then K is an ellipsoid. As a part of the proof, smoothness p…