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

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 König 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. For the four-dimensional cube, we further use these formulas to identify candidate global maxima and minima for slice and slab volumes; the candidates are verified to be critical points by exact symbolic computation and are checked numerically to give the global extrema.