A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

TL;DR

This paper establishes that the normalized volume of the free sum of full-dimensional lattice polytopes equals the product of their individual normalized volumes, revealing a simple multiplicative relationship.

Contribution

It provides a precise formula linking the normalized volume of free sums to the volumes of the summands, a new insight in convex geometry.

Findings

Normalized volume of free sum equals product of summands' volumes

Valid for full-dimensional lattice polytopes

Simplifies volume calculations for free sums

Abstract

The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.