Matheron's Conjecture for the Covariogram Problem

The covariogram of a convex body K provides the volumes of the intersections of K with all its possible translates. Matheron conjectured in 1986 that this information determines K among all convex bodies, up to certain known ambiguities. It is proved that this is the case if K ⊂ R 2 is not C 1 , or it is not strictly convex, or its boundary contains two arbitrarily small C 2 open portions ‘on opposite sides’. Examples are also constructed that show that this conjecture is false in R n for any n ⩾ 4.