Parallelepipeds of maximal facet area in ellipsoids, through prescribed boundary points
Authors
Tomasz Kania
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.
