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 ⊂ R2 is not C1, or it is not strictly convex, or its boundary contains two arbitrarily small C2 open portions ‘on opposite sides’. Examples are also constructed that show that this conjecture is false in Rn for any n 4.