Aleksandrov projection problem for convex lattice sets

TL;DR

This paper investigates a discrete version of the Aleksandrov projection problem for convex lattice sets, proving that under certain conditions in two dimensions, the sets are identical.

Contribution

It provides a positive solution to the discrete Aleksandrov projection problem in two dimensions with an additional hypothesis.

Findings

Sets are equal if their projections have equal lattice points under specified conditions.

The result applies specifically to convex integer polytopes in z2.

Additional hypothesis involving doubled sets is necessary for the conclusion.

Abstract

Let $K$ and $L$ be origin-symmetric convex integer polytopes in $\mathbb{R}^n$. We study a discrete analogue of the Aleksandrov projection problem. If for every $u\in \mathbb{Z}^n$, the sets $(K\cap \mathbb{Z}^n)|u^\perp$ and $(L\cap \mathbb{Z}^n)|u^\perp$ have the same number of points, is then $K=L$? We give a positive answer to this problem in $\mathbb{Z}^2$ under an additional hypothesis that $(2K\cap \mathbb{Z}^2)|u^\perp$ and $(2L\cap \mathbb{Z}^2)|u^\perp$ have the same number of points.