Quantum jumps of normal polytopes

TL;DR

This paper introduces a new partial order on normal polytopes in R^d, explores its complex structure, and presents the first examples of maximal elements in certain dimensions, combining theory, random generation, and computer search.

Contribution

It defines a novel partial order on normal polytopes, derives bounds on elementary relations, and provides the first known examples of maximal polytopes in specific dimensions.

Findings

Established a partial order on normal polytopes with rich combinatorial structure

Derived arithmetic bounds on quantum jumps between polytopes

Discovered the first examples of maximal normal polytopes in dimensions 4 and 5

Abstract

We introduce a partial order on the set of all normal polytopes in R^d. This poset NPol(d) is a natural discrete counterpart of the continuum of convex compact sets in R^d, ordered by inclusion, and exhibits a remarkably rich combinatorial structure. We derive various arithmetic bounds on elementary relations in NPol(d), called "quantum jumps". The existence of extremal objects in NPol(d) is a challenge of number theoretical flavor, leading to interesting classes of normal polytopes: minimal, maximal, spherical. Minimal elements in NPol(5) have played a critical role in disproving various covering conjectures for normal polytopes in the 1990s. Here we report on the first examples of maximal elements in NPol(4) and NPol(5), found by a combination of the developed theory, random generation, and extensive computer search.