Parallelepipeds of maximal facet area and total edge length in ellipsoids, through prescribed boundary points

Abstract

Let \[ \mathcal{E}_A=\{x\in\mathbb{R}^n:x^{\top}A^{-1}x\le 1\},\qquad n\ge2, \] where $A$ is real symmetric positive definite. We study full-dimensional parallelepipeds whose $2^n$ vertices lie on $\partial {\mathcal{E}}_A$. First we show that such parallelepipeds are necessarily centred at the origin and are precisely the images, under $A^{1/2}$, of orthotopes inscribed in the Euclidean unit sphere. This reduces the extremal questions to finite-dimensional linear algebra. For the total length $L$ of the one-skeleton we prove \[ L_{\max}(\mathcal{E}_A)=2^n\sqrt{\operatorname{tr} A}. \] Moreover, the prescribed-vertex problem for $L$ has the same answer in every dimension: for every $x_0\in \partial {\mathcal{E}}_A$ there is an inscribed parallelepiped with vertex $x_0$ and total edge length $2^n\sqrt{\mathrm{tr}A}$. The proof uses the Schur--Horn theorem applied to the trace-zero matrix $A-\mathrm{tr}(A)y_0{y}_0^{\top }$, where $y_0=A^{-1/2}{x}_0$. For the total $(n-1)$-dimensional measure $S$ of the facets we prove \[ S_{\max}(\mathcal{E}_A)=2^n n^{-(n-2)/2}\sqrt{\det A}\,\sqrt{\operatorname{tr}(A^{-1})}. \] For $n\ge 3$ the maximisers are more rigid: on the sphere they are orthotopes with all edge lengths equal and with a Schur--Horn equal diagonal condition for $A^{-1}$. The prescribed-vertex facet-area problem is therefore equivalent to a restricted Schur--Horn problem with a prescribed barycentric basis. In dimension two this recovers the Connes--Zagier property for ellipses. In dimension three, however, the direct higher-dimensional analogue fails for triaxial ellipsoids at principal-axis vertices; an exact obstruction is given.

8 pp., accepted for publication in Results in Mathematics