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

Marie-Charlotte Brandenburg, Jesús A. De Loera, Yu Luo, Chiara Meroni

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.

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 -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.

Paper Structure

This paper contains 36 sections, 8 theorems, 64 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Algorithm for moments of slices and slabs of polyhedral norm balls.
  4. The complete family of volume formulas of slices and slabs of the $4$-cube.
  5. Higher moments and critical-point computations in low dimensions.
  6. Organization.
  7. Methodology
  8. Equivalence Classes of Slices via Vertex Ordering
  9. From Slices to Slabs
  10. Vertices of a slab
  11. Triangulation computation
  12. Volume and Higher Moment Formulas
  13. Proof of \ref{['thm:main']}
  14. Algebraic analysis to recover critical points of moments
  15. Slices and Slabs of Low-Dimensional Cubes
  16. ...and 21 more sections

Key Result

Theorem 1

Fix $d\in\mathbb{N}$ and a polyhedral norm $\| \cdot \|$ on $\mathbb{R}^d$. For any $M \in \mathbb Z_{\geq 0}$, the algebraic method introduced in sec:methodology computes the $M$-th moments $\int_{\mathop{\mathrm{slice}}\nolimits(a,t,\|\cdot\|)}\ \sum_{i=1}^d x_i^M$ and $\int_{\mathop{\mathrm{slab}

Figures (5)

  • Figure 1: Left: $\operatorname{slab}((\frac{2}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}}),\frac{\sqrt{6}}{2},B_{\infty}^3)$. Right: its barycentric triangulation, from \ref{['ex:triangulation']}.
  • Figure 2: Center: hyperplane sweep across $B_{\|\cdot\|_\infty}$ in $\mathbb{R}^2$. Left: quadrilateral slab associated to $(a_{\mathrm{rep}},t_{\mathrm{rep}})\in C_{1,1}$. Right: hexagonal slab associated to $(a_{\mathrm{rep}},t_{\mathrm{rep}})\in C_{1,2}$. The rational functions describing the volumes of these two types of slabs are distinct and are computed in \ref{['ex:2cube_2moment']}.
  • Figure 3: Left: The two fundamental open slicing chambers of the two-dimensional cube, displayed in coordinates $(\alpha,t)$, where $(a_1,a_2) = (\cos \alpha, \sin \alpha)$. Right: The piecewise rational function $f = \{f_{1}^{(2,2)},f_{2}^{(2,2)}\}$ of the second moment from \ref{['ex:algebraic_2cube_2moment']}, together with its maxima (black curve) for each value of $t$.
  • Figure 4: The function $f(\alpha,t_{\mathrm{rep}}) = \{f_{1}^{(2,2)}(\alpha,t_{\mathrm{rep}}),f_{2}^{(2,2)}(\alpha,t_{\mathrm{rep}})\}$ of the second moment of the two-dimensional cube from \ref{['ex:algebraic_2cube_2moment']}, where $(a_1,a_2) = (\cos \alpha, \sin \alpha)$ and $t_{\mathrm{rep}} = 0.2, 0.8, 1.2$ from left to right. The green part of each curve is its restriction to $C_{1,1}$, and the red part is its restriction to $C_{1,2}$. Only in the central figure is there a maximum in the interior of a slicing chamber.
  • Figure 5: A representative for each maximal slicing chamber of the square, up to symmetry. Left: $(a,t)\in C_{1,1}$. Right: $(a,t)\in C_{1,2}$.

Theorems & Definitions (25)

  • Theorem 1: Polynomial-time moment computation
  • Theorem 2: Volume formulas of the 4-dimensional cube
  • Theorem 3: Algebraic critical points of moments of square slices
  • Theorem 4: Moments of slices and slabs
  • Definition 2.1: Sweep arrangement
  • Definition 2.2: Maximal Chamber, slices
  • Lemma 5: slices
  • Example 1
  • Lemma 6
  • proof
  • ...and 15 more