The boundary volume of a lattice polytope

TL;DR

This paper derives a formula for the boundary volume of lattice polytopes using boundary lattice points, providing new characterizations for reflexive polytopes and formulas for their f-vectors in low dimensions.

Contribution

It introduces a novel formula for boundary volume based on boundary lattice points and applies it to characterize reflexive polytopes and compute f-vectors in dimensions 3 to 5.

Findings

Boundary volume formula in terms of boundary lattice points

Necessary and sufficient condition for reflexivity

Explicit f-vector formulas for dimensions 3, 4, and 5

Abstract

For a d-dimensional convex lattice polytope P, a formula for the boundary volume 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.