On the reconstruction of planar lattice-convex sets from the covariogram

TL;DR

This paper investigates the problem of reconstructing planar lattice-convex sets from their covariogram, providing conditions under which reconstruction is possible and extending known counterexamples.

Contribution

It offers a partial positive solution for 2D cases and extends the class of known counterexamples for reconstructibility of lattice-convex sets.

Findings

Covariogram determines lattice-convex sets up to translation and reflection in 2D under mild assumptions.

Extended the set of counterexamples to include an infinite family of non-reconstructible sets.

Provided insights into the limitations of reconstructing lattice-convex sets from covariogram data.

Abstract

A finite subset $K$ of $\mathbb{Z}^d$ is said to be lattice-convex if $K$ is the intersection of $\mathbb{Z}^d$ with a convex set. The covariogram $g_K$ of $K\subseteq \mathbb{Z}^d$ is the function associating to each $u \in \integer^d$ the cardinality of $K\cap (K+u)$. Daurat, G\'erard, and Nivat and independently Gardner, Gronchi, and Zong raised the problem on the reconstruction of lattice-convex sets $K$ from $g_K$. We provide a partial positive answer to this problem by showing that for $d=2$ and under mild extra assumptions, $g_K$ determines $K$ up to translations and reflections. As a complement to the theorem on reconstruction we also extend the known counterexamples (i.e., planar lattice-convex sets which are not reconstructible, up to translations and reflections) to an infinite family of counterexamples.