On Set Covariance and Three-Point Test Sets

Abstract

The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of \(\mathbb{R}^d \)is studied. In particular, the connection of the first order directional derivatives of the described characteristic with the oriented and the unoriented normal measure of a set representable as a finite union of sets with positive reach is established. For smooth convex bodies with positive curvatures, the second and the third order directional derivatives of the characteristic is computed.