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

Let $${\mathcal{E}}_A=\{x\in {\mathbb{R}}^n:x^{\mathrm{\top }}{A}^{-1}x\le 1\},n\ge 2,$$ where $A$ is real symmetric positive definite. We study full-dimensional parallelepipeds whose $2^n$ vertices lie on $\mathrm{\partial }{\mathcal{E}}_A$. First we show that such parallelepipeds are necessarily centred at the origin and are precisely the images, under $A^{1\mathrm{/}2}$, 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 $$L_{\max }({\mathcal{E}}_A)=2^n\sqrt{\mathrm{tr}A}\mathrm{.}$$ Moreover, the prescribed-vertex problem for $L$ has the same answer in every dimension: for every $x_0\in \mathrm{\partial }{\mathcal{E}}_A$ 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 $A-\mathrm{tr}(A)y_0{y}_0^{\mathrm{\top }}$, where $y_0=A^{-1\mathrm{/}2}{x}_0$. For the total $(n-1)$-dimensional measure $S$ of the facets we prove $$S_{\max }({\mathcal{E}}_A)=2^n{n}^{-(n-2)\mathrm{/}2}\sqrt{\det A}\sqrt{\mathrm{tr}(A^{-1})}\mathrm{.}$$ 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.