On the Statistics of Lattice Polytopes

TL;DR

This paper introduces probability measures for lattice polytopes using reflexivity concepts, exploring their statistical properties, applications in discrete geometry, and implications for algebraic geometry and string theory.

Contribution

It develops a framework for analyzing the statistical properties of lattice polytopes, including reflexivity and IP-confined polytopes, with new results on 3D IP-simplices.

Findings

Analysis of randomness in self-duality of reflexive polytopes

Implications for expected counts of polytopes in higher dimensions

List of 3D IP-simplices not IP-confined

Abstract

We use the notions of reflexivity and of reflexive dimensions in order to introduce probability measures for lattice polytopes and initiate the investigation of their statistical properties. Examples of applications to discrete geometry include a study of randomness of self-duality of reflexive polytopes and implications for expectation values of the numbers of such polytopes in higher dimensions. We also discuss enumeration problems and related algorithms and point out interesting open problems. In this context we define the notion of IP-confined polytopes. Our new results include the list of IP-simplices in 3 dimensions that are not IP-confined. The main motivation for the study of these issues comes from applications in algebraic geometry and string theory.