Unique Determination of Convex Lattice Sets

Abstract

Let K and L be origin-symmetric convex lattice sets in $$\mathbb Z^n$$Zn. We study a discrete analogue of the Aleksandrov theorem for the surface areas of projections. If for every $$u\in \mathbb Z^n$$uźZn, the sets $$(K|u^\perp )\cap \partial (\hbox {conv}(K)|u^\perp )$$(K|uź)źź(conv(K)|uź) and $$(L|u^\perp )\cap \partial (\hbox {conv}(L)|u^\perp )$$(L|uź)źź(conv(L)|uź) have the same number of points, is then necessarily $$K=L$$K=L? We give a positive answer to this question in $$\mathbb Z^3$$Z3. In higher dimensions, we obtain an analogous result when $$\hbox {conv}(K)$$conv(K) and $$\hbox {conv}(L)$$conv(L) are zonotopes.