Combinatorial mixed valuations

TL;DR

This paper introduces combinatorial mixed valuations for translation-invariant valuations on polytopes, showing they often inherit properties like monotonicity, which has implications for computational geometry and extends previous work on discrete volumes.

Contribution

It defines combinatorial mixed valuations, explores their properties, and connects them to existing concepts like monotonicity and discrete volumes, providing new insights and generalizations.

Findings

Combinatorial mixed valuations are often monotone and nonnegative.

For rational polytopes, combinatorial mixed monotonicity is equivalent to monotonicity.

The results generalize and strengthen previous work on discrete mixed volumes.

Abstract

Combinatorial mixed valuations associated to translation-invariant valuations on polytopes are introduced. In contrast to the construction of mixed valuations via polarization, combinatorial mixed valuations reflect and often inherit properties of inhomogeneous valuations. In particular, it is shown that under mild assumptions combinatorial mixed valuations are monotone and hence nonnegative. For combinatorially positive valuations, this has strong computational implications. Applied to the discrete volume, the results generalize and strengthen work of Bihan (2015) on discrete mixed volumes. For rational polytopes, it is proved that combinatorial mixed monotonicity is equivalent to monotonicity. Stronger even, a conjecture is substantiated that combinatorial mixed monotonicity implies the homogeneous monotonicity in the sense of Bernig--Fu (2011).