Unique Determination of Convex Lattice Sets

Abstract

Let K and L be origin-symmetric convex lattice sets in $\mathbb Z^n$. We study a discrete analogue of the Aleksandrov theorem for the surface areas of projections. If for every $u\in \mathbb Z^n$, the sets $(K|u^\perp )\cap \partial (\hbox {conv}(K)|u^\perp )$ and $(L|u^\perp )\cap \partial (\hbox {conv}(L)|u^\perp )$ have the same number of points, is then necessarily $K=L$? We give a positive answer to this question in $\mathbb Z^3$. In higher dimensions, we obtain an analogous result when $\hbox {conv}(K)$ and $\hbox {conv}(L)$ are zonotopes.