A short, based on the mixed volume, proof of Liggett's theorem on the convolution of ultra-logconcave sequences

TL;DR

This paper offers a concise proof of Liggett's theorem on the convolution of ultra-logconcave sequences, utilizing mixed volume concepts from convex geometry, simplifying the original proof.

Contribution

The paper introduces a novel, shorter proof of Liggett's theorem using mixed volume techniques, enhancing understanding of ultra-logconcavity preservation under convolution.

Findings

Proof of Liggett's theorem is shorter and more geometric.

Confirms convolution preserves ultra-logconcavity.

Connects convex geometry with sequence log-concavity.

Abstract

R. Pemantle conjectured, and T.M. Liggett proved in 1997, that the convolution of two ultra-logconcave is ultra-logconcave. Liggett's proof is elementary but long. We present here a short proof, based on the mixed volume of convex sets.