Minkowski valuations on lattice polytopes

TL;DR

This paper classifies Minkowski valuations on lattice polytopes that are invariant under certain group actions, revealing that they are essentially multiples of known geometric constructs or combinations involving a new discrete Steiner point.

Contribution

It provides a complete classification of specific Minkowski valuations on lattice polytopes, identifying the only possible forms under group invariance and translation invariance.

Findings

Contravariant valuations are multiples of projection bodies.

Equivariant valuations are generalized difference bodies plus multiples of a new discrete Steiner point.

The work introduces a new discrete Steiner point concept.

Abstract

A complete classification is established of Minkowski valuations on lattice polytopes that intertwine the special linear group over the integers and are translation invariant. In the contravariant case, the only such valuations are multiples of projection bodies. In the equivariant case, the only such valuations are generalized difference bodies combined with multiples of the newly defined discrete Steiner point.