Unimodular triangulation and bounds for smooth Fano polytopes

TL;DR

This paper constructs a specific unimodular triangulation for simplicial smooth Fano polytopes, proves their delta-vectors are unimodal, and establishes bounds on their volume.

Contribution

It provides a concrete triangulation method, proves unimodality of delta-vectors, and derives volume bounds for these polytopes, advancing understanding of their combinatorial properties.

Findings

Established a concrete unimodular triangulation for simplicial smooth Fano polytopes.

Proved the delta-vector of such polytopes is unimodal.

Derived upper and lower bounds for the volume of these polytopes.

Abstract

Let $P$ be a simplicial smooth Fano polytope. We provide a concrete unimodular triangulation of $P$. We prove that the delta-vector of a simplicial smooth Fano polytope is unimodal and we give upper and lower bound for the volume of simplicial smooth Fano polytopes.