Critical moments of slices and slabs of the cube (and other polyhedral norms)

TL;DR

This paper introduces a unified algebraic-combinatorial framework for computing explicit formulas for volumes and moments of slices and slabs of polyhedral norm balls, extending previous work and providing new computational tools and formulas.

Contribution

It develops a polynomial-time algorithm for fixed dimensions, extends methods to slabs, and offers explicit formulas and critical point analysis for slices and slabs of cubes.

Findings

Recovered known volume formulas for 2D and 3D cubes

Identified a new family of formulas for 4D cube slices and slabs

Computed explicit higher moments for 2D and 3D cubes

Abstract

In this article, we present a unified algebraic-combinatorial framework for computing explicit, piecewise rational, and combinatorially indexed parametric formulas for volumes and higher moments of slices and slabs of polyhedral norm balls. Our main method builds on prior work concerning a combinatorial decomposition of the parameter space of all slices of a polytope. We extend this framework to slabs, and find a polynomial-time algorithm in fixed dimension. We also exhibit computational methods to obtain moments of arbitrary order for all slices or slabs of any polyhedral norm ball, and an algebraic framework for analyzing their critical points. In addition, we present an experimental study of the $d$-dimensional unit cube. Our analysis recovers and reinterprets the known volume formulas for slabs and slices of the two- and three-dimensional cubes, first obtained by K\"onig and Koldobsky. Moreover, our method identifies a new complete family of fourteen rational functions giving the volumes of slices and slabs of the four-dimensional cube. We further compute explicit higher moments of slices and slabs in dimensions two and three, and derive explicit formulas for moments of arbitrary order for slices of the two-dimensional cube, describing their critical points.