Covariogram of non-convex sets

TL;DR

This paper investigates the covariogram function of sets in the plane, extending known uniqueness results from convex bodies to broader classes, and establishes limitations for discrete covariogram equivalence among polyominoes.

Contribution

It extends covariogram determination results beyond convex bodies and shows non-congruence of small planar polyominoes with identical discrete covariograms.

Findings

Covariogram determines certain non-convex planar sets.

No two non-congruent planar polyominoes with fewer than 9 points share the same discrete covariogram.

Abstract

The covariogram of a compact set A contained in R^n is the function that to each x in R^n associates the volume of A intersected with (A+x). Recently it has been proved that the covariogram determines any planar convex body, in the class of all convex bodies. We extend the class of sets in which a planar convex body is determined by its covariogram. Moreover, we prove that there is no pair of non-congruent planar polyominoes consisting of less than 9 points that have equal discrete covariogram.