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Parallelepipeds of maximal facet area in ellipsoids, through prescribed boundary points

Tomasz Kania

TL;DR

The paper determines sharp global bounds for two extremal functionals of parallelepipeds inscribed in an ellipsoid: the total facet area $S(P)$ and the total edge length $L(P)$. It reduces the problem to the unit sphere via $E= B S^{n-1}$ and analyzes Gram-structure alongside Schur–Horn inequalities to identify maximisers, showing $S_{ ext{max}}(E)=2^{n}n^{-(n-2)/2}\sqrt{\det A}\sqrt{\operatorname{tr}(A^{-1})}$ and $L_{ ext{max}}(E)=2^{n}\sqrt{\operatorname{tr}(A)}$, with attainments by inscribed images of orthotopes. The work also addresses vertex-constrained variants, proving a universal bound in general and complete planar results (and principal-axis cases) in higher dimensions, including explicit equalisation constructions. These results generalize classical planar findings for ellipses (Richard; Connes–Zagier) to the $n$-dimensional setting and connect geometric extremality with matrix-analytic tools.

Abstract

Let $E=\{x\in\mathbb{R}^n : x^{\top}A^{-1}x=1\}$ be an $n$-dimensional ellipsoid with $A$ positive definite. Among all $n$-dimensional parallelepipeds inscribed in $E$ (all $2^n$ vertices on $\partial E$), we study (i) the total $(n-1)$-dimensional measure of the facets (`surface area'), and (ii) the total $1$-dimensional measure of the $1$-skeleton (`circumference' or total edge length). We prove that the sharp global bounds $$ S_{\max}(E)=2^{n}n^{-(n-2)/2}\sqrt{\det A}\;\sqrt{\operatorname{tr}(A^{-1})}, \qquad L_{\max}(E)=2^{n}\sqrt{\operatorname{tr}(A)}. $$ are attained by inscribed images of orthotopes in the unit sphere; we give another proof in dimension $n=2$, originally proved by Richard, and, for $n\geqslant 3$, we prove such a maximisation result in the principal-axis case. In that case, we provide a constructive diagonal equalisation argument that preserves the barycentric constraint.

Parallelepipeds of maximal facet area in ellipsoids, through prescribed boundary points

TL;DR

The paper determines sharp global bounds for two extremal functionals of parallelepipeds inscribed in an ellipsoid: the total facet area and the total edge length . It reduces the problem to the unit sphere via and analyzes Gram-structure alongside Schur–Horn inequalities to identify maximisers, showing and , with attainments by inscribed images of orthotopes. The work also addresses vertex-constrained variants, proving a universal bound in general and complete planar results (and principal-axis cases) in higher dimensions, including explicit equalisation constructions. These results generalize classical planar findings for ellipses (Richard; Connes–Zagier) to the -dimensional setting and connect geometric extremality with matrix-analytic tools.

Abstract

Let be an -dimensional ellipsoid with positive definite. Among all -dimensional parallelepipeds inscribed in (all vertices on ), we study (i) the total -dimensional measure of the facets (`surface area'), and (ii) the total -dimensional measure of the -skeleton (`circumference' or total edge length). We prove that the sharp global bounds are attained by inscribed images of orthotopes in the unit sphere; we give another proof in dimension , originally proved by Richard, and, for , we prove such a maximisation result in the principal-axis case. In that case, we provide a constructive diagonal equalisation argument that preserves the barycentric constraint.
Paper Structure (6 sections, 14 theorems, 58 equations)

This paper contains 6 sections, 14 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Spherical reductions
  3. Maximising the 1-dimensional measure (total edge length)
  4. Maximising the (n-1)-dimensional measure (hypersurface measure)
  5. A prescribed vertex: sharp bound and attainment in low dimension
  6. Further remarks

Key Result

Lemma 2.1

Let $P$ be inscribed in $E$ and set $Q:=B^{-1}P$. Then $Q$ is inscribed in the unit sphere $S^{n-1}$. Conversely, if $Q\subset\mathbb{R}^n$ is inscribed in $S^{n-1}$, then $P:=BQ$ is inscribed in $E$. Moreover, $Q\subset S^{n-1}$ is an inscribed parallelepiped if and only if its edge vectors $w_1,\d where $u_1,\dots,u_n$ is an orthonormal basis of $\mathbb{R}^n$. In particular, $Q$ is an orthotope

Theorems & Definitions (31)

  • Lemma 2.1
  • proof
  • Theorem 3.1: Maximal total edge length
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • Remark 3.5
  • Proposition 4.1
  • ...and 21 more