Matheron's Conjecture for the Covariogram Problem

The covariogram of a convex body K provides the volumes of theintersections of K with all its possible translates. Matheronconjectured in 1986 that this information determines K amongall convex bodies, up to certain known ambiguities. It is provedthat this is the case if K R² is not C¹, or it is not strictlyconvex, or its boundary contains two arbitrarily small C² openportions ‘on opposite sides’. Examples are alsoconstructed that show that this conjecture is false in Rⁿ forany n 4.