Normal polytopes: between discrete, continuous, and random

TL;DR

This survey explores the interplay between discrete, continuous, and random perspectives on normal polytopes, presenting new results, conjectures, and outlining future challenges in the broader context of convex polytopes.

Contribution

It provides an updated, expanded overview of normal polytopes, introduces new conjectures, and discusses future research directions in the field.

Findings

New results on normal polytopes

Proposed conjectures on their interactions

Outlined challenges in convex polytope study

Abstract

The first three sections of this survey represent an updated and much expanded version of the abstract of my talk at FPSAC'2010: new results are incorporated and several concrete conjectures on the interactions between the three perspectives on normal polytopes in the title are proposed. The last section outlines new challenges in general convex polytopes, motivated by the study of normal polytopes.