Combinatorial Methods for Minkowski Tensors of Polytopes

TL;DR

This paper develops a generating function approach to compute Minkowski surface tensors of polytopes, extending previous methods and providing explicit formulas for simplicial cases.

Contribution

It introduces a novel generating function framework for Minkowski tensors and extends the adjoint polynomial concept to polytope boundaries, with explicit formulas for simplicial polytopes.

Findings

Explicit formulas for surface tensors of simplicial polytopes

Extension of the adjoint polynomial to boundary complexes

A new generating function method for Minkowski tensor calculations

Abstract

In this paper we use a generating function approach to record and calculate entries of the Minkowski tensors of a polytope. We focus on ''surface tensors'', extending the methods used in arXiv:1807.10258 for moments of the uniform distribution which correspond to volume tensors. In this context we also extend the definition of the adjoint polynomial to the boundary complex of a polytope with simplicial facets. In the case of simplicial polytopes we give an explicit formulation for these surface tensors.