Facets and volume of Gorenstein Fano polytopes

TL;DR

This paper introduces new classes of normal Gorenstein Fano polytopes, showing that order and chain polytopes are faces of such polytopes, and explores their combinatorial properties like Ehrhart polynomials and volume.

Contribution

It provides explicit constructions of normal Gorenstein Fano polytopes containing order and chain polytopes as faces, expanding understanding of their structure and properties.

Findings

Order and chain polytopes are faces of higher-dimensional Gorenstein Fano polytopes.

Analysis of Ehrhart polynomials and volume of the constructed polytopes.

Discovery of interesting examples of Gorenstein Fano polytopes.

Abstract

It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension $d$ is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension $d + 1$. Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.