Aleksandrov projection problem for convex lattice sets

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.