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.
Categories: math.CA