Tensor valuations on lattice polytopes

Abstract

The Ehrhart polynomial and the reciprocity theorems by Ehrhart & Macdonald are extended to tensor valuations on lattice polytopes. A complete classification is established of tensor valuations of rank up to eight that are equivariant with respect to the special linear group over the integers and translation covariant. Every such valuation is a linear combination of the Ehrhart tensors which is shown to no longer hold true for rank nine. 2010

Ehrhart tensor polynomials

“…A unimodular transformation of a polytope P ∈ P(Z d ) is a GL(Z d ) transformation of P paired with a translation. Similar to McMullen, a reciprocity theorem was given for translation covariant valuations in [26]. Extending the classical Ehrhart-Macdonald reciprocity, the following reciprocity theorem gives the special case of the discrete moment tensor.…”

“…For r = 1, L 1 (P ) equals the discrete moment vector defined in [9]. Based on results by Khovanskiȋ and Pukhlikov [31] and Alesker [1], it was identified in [26] that L r (nP ) is given by a polynomial, for any n ∈ N, extending Ehrhart's celebrated result for the lattice point enumerator [13].…”

“…Recently, Ludwig and Silverstein [26] introduced Ehrhart tensor polynomials based on discrete moment tensors that were defined by Böröczky and Ludwig [10]. Let P(Z d ) denote the family of convex polytopes with vertices in Z d called lattice polytopes and let T r be the vector space of symmetric tensors of rank r on R d .…”

“…The only coefficients that are known to have explicit geometric descriptions are the leading, second-highest, and constant coefficients for the classic Ehrhart polynomial (see, e.g., [6]). For the Ehrhart tensor polynomial, the leading and constant coefficients were given in [26] and we give an interpretation for the second-highest coefficient (Proposition 2.3) as the weighted sum of moment tensors over the facets of the polytope; the descriptions of all are given in Section 2.…”

SL(m,C)-equivariant and translation covariant continuous tensor valuations

“…In these cases, the action of a group G acting both on K(V) and A is also considered and usually a characterization result for different groups G and actions is studied. The related problem of tensor valuations on lattice polytopes is discussed in the pioneering paper of Ludwig and Silverstein [43].…”