The boundary volume of a lattice polytope

Abstract

For a $d$-dimensional convex lattice polytope $P$, a formula for the boundary volume \vol{\partial P} is derived in terms of the number of boundary lattice points on the first \floor{d/2} dilations of $P$. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulae for the $f$-vector of a smooth polytope in dimensions $3$, $4$, and $5$. We also give applications to reflexive order polytopes, and to the Birkhoff polytope. DOI: 10.1017/S000497271100257