Reverse Faber–Krahn conjecture for origin-symmetric convex bodies

Let $K ot i 0$ be an origin-symmetric convex body in $\mathbb R^m$, meaning that $x\in K$ if and only if $-x\in K$. Write $\lambda(K)$ for the first Dirichlet eigenvalue of $K$, and let $\mathrm{vol}_m$ denote Lebesgue measure. Reverse Faber–Krahn conjecture. For every origin-symmetric convex body $K\subseteq\mathbb R^m$, there is a volume-preserving linear transformation $S\in\mathsf{SL}_m(\mathbb R)$ such that

$$\lambda(SK)\mathrm{vol}_m(K)^{2/m}\asymp m.$$

The classical Faber–Krahn inequality identifies the Euclidean ball as the minimizer of the corresponding normalized eigenvalue, whereas this conjecture asserts that every origin-symmetric convex body can be linearly positioned so that the normalized quantity has the Euclidean order of magnitude. The paper proves this phenomenon for unit balls of unitarily invariant matrix norms.