Facets and volume 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.

Enriched order polytopes and enriched Hibi rings

“…In each dimension there exist only finitely many reflexive polytopes up to unimodular equivalence [13] and all of them are known up to dimension 4 [12]. Recently, several classes of reflexive polytopes were constructed by an algebraic technique on Gröbner bases, cf., [10,11,15]. The algebraic technique is based on the following lemma that follows from the argument in [ In order to use Lemma 3.1 for enriched chain polytopes C (e) P , we study the toric ideal of C (e) P .…”

Reflexive polytopes arising from perfect graphs

“…• Reflexive polytopes arising from the order polytopes and the chain polytopes of finite partially ordered sets ( [11,13,14,15]). • Reflexive polytopes arising from the stable sets polytopes of perfect graphs ( [21]).…”

Reflexive polytopes arising from partially ordered sets and perfect graphs

“…Recently, the study on reflexive polytopes with the integer decomposition property has been achieved in the frame of Gröbner bases [11,[13][14][15][16]21]. First, we recall the definition of a reflexive polytope with the integer decomposition property.…”

“…By using these constructions, we can obtain several classes of reflexive polytopes with the integer decomposition property. In fact, in [11,[13][14][15], large classes of reflexive polytopes with the integer decomposition property which arise from finite partially ordered sets are given. Moreover, in [16,21], large classes of reflexive polytopes with the integer decomposition property which arise from perfect graphs are given.…”

“…Moreover, in [16,21], large classes of reflexive polytopes with the integer decomposition property which arise from perfect graphs are given. In particular, we focus on the following result: Theorem 1.1 ( [13,15]) Let P and Q be finite partially ordered sets on [d] = {1, . .…”