The separation modulus of unitarily invariant matrix norms

Abstract

If $X=(M_n(\mathbb{R}),\|\cdot\|)$ is a unitarily invariant normed space on , then we prove (via exact computations for a Jacobi orthogonal random matrix ensemble) that the spectral gap of the Laplacian with Dirichlet boundary conditions on the unit ball $B_X$ of $X$ satisfies $\lambda(X)\asymp n^3 \|I\|^2$. This leads to a confirmation of the weak isomorphic reverse isoperimetry conjecture for $X$, namely, we demonstrate that there exists a convex body $L=L_X\subset B_X$ such that $\mathrm{vol}_{n^2}(L)^{1/n^2}\asymp \mathrm{vol}_{n^2}(B_{X})^{1/n^2}$, yet its isoperimetric quotient is at most a universal constant multiple of $n$. As a corollary (and motivation) of these results, we deduce that the separation modulus of $X$ satisfies $\mathsf{SEP}(X)\asymp \sqrt{n}\|I_n\|_X\mathrm{diam}(B_X)$, where $\mathrm{diam}(B_X)$ is the diameter of $B_X$ with respect to the standard Euclidean metric on $M_n(\mathbb{R})$. Assuming oracle access to norm evaluations in $X$, by combining this with a new deterministic algorithm for efficiently computing a $O(1)$-approximation of the diameter of convex bodies in $\mathbb{R}^n$ that are given by a weak membership oracle and are symmetric with respect to coordinate permutations and reflections about the axes, we obtain an oracle polynomial time algorithm whose output is guaranteed to be the separation modulus of $X$ up to positive universal constant factors. We also deduce an upper bound on the Lipschitz extension modulus of $X$ that improves over the previously best-known bound even in the special case when $X$ is $M_n(\mathbb{R})$ equipped with the $\ell_{2}^n\to \ell_{2}^n$ operator norm.