Convexity in cristallographical lattices

Abstract

A subset of a (cristallographical) lattice ℒⁿ is called convex whenever it is the intersection of the lattice with a convex set of the affine space containing ℒⁿ. We give a characterization of the convex sets which is intrinsic to the lattice and do the same for other related notions, e.g. the boundary of a convex set of ℒⁿ. A statement analogous to Helly's theorem is also proved.