Parallelepipeds of Maximal Facet Area and Total Edge Length in Ellipsoids, Through Prescribed Boundary Points

Let \[ E_A=\{x^n:x^{}A⁻¹x≤ 1\}, n≥2, \] where A is real symmetric positive definite. We study full-dimensional parallelepipeds whose 2ⁿ vertices lie on A. First we show that such parallelepipeds are necessarily centred at the origin and are precisely the images, under A1/2, 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 \[ Lₘₐₓ(E_A)=2^n√tr A. \] Moreover, the prescribed-vertex problem for L has the same answer in every dimension: for every x₀∈A there is an inscribed parallelepiped with vertex x₀ and total edge length 2ⁿ√tr A. The proof uses the Schur--Horn theorem applied to the trace-zero matrix A-tr(A)y₀y₀^, where y₀=A-1/2x₀. For the total $(n-1)$-dimensional measure S of the facets we prove \[ Sₘₐₓ(E_A)=2^n n-(n-2)/2√ A\,√{tr(A⁻¹)}. \] For n≥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.