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Oda's conjecture and bounds for smooth Fano polytopes

Gábor Hegedüs

Abstract

Oda asked the following prominent question in Oberwolfbach: Let $P$ be a smooth lattice polytope. Does $P$ have the integer decomposition property? We answer Oda's question in the affirmative in the special case of smooth Fano polytopes. We prove that the delta-vector of a smooth Fano polytope is unimodal and we give upper and lower bound for the volume of smooth Fano polytopes.

Oda's conjecture and bounds for smooth Fano polytopes

Abstract

Oda asked the following prominent question in Oberwolfbach: Let be a smooth lattice polytope. Does have the integer decomposition property? We answer Oda's question in the affirmative in the special case of smooth Fano polytopes. We prove that the delta-vector of a smooth Fano polytope is unimodal and we give upper and lower bound for the volume of smooth Fano polytopes.
Paper Structure (5 sections, 18 theorems, 18 equations)

This paper contains 5 sections, 18 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Combinatorics of polytopes
  3. Reflexive polytopes
  4. Proofs of bounds for smooth polytopes
  5. Concluding remarks

Key Result

Theorem 1.2

Let $P$ be a $d$-dimensional convex lattice polytope with $0\in P^\circ$, where $P^\circ$ denote the strict interior of $P$. The following statements are equivalent:

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Proposition 1.9
  • Corollary 1.10
  • ...and 16 more