Weak isomorphic reverse isoperimetry conjecture

Let $K\subseteq\mathbb R^m$ be an origin-symmetric convex body, meaning that $x\in K$ if and only if $-x\in K$. Let $\mathsf{SL}_m(\mathbb R)$ denote the group of volume-preserving linear transformations, and let $\mathrm{iq}(L)$ denote the isoperimetric quotient of a convex body $L$. Weak isomorphic reverse isoperimetry conjecture. There exist $S\in\mathsf{SL}_m(\mathbb R)$ and an origin-symmetric convex body $L\subseteq SK$ such that

$$\mathrm{vol}_m(L)^{1/m}\gtrsim \mathrm{vol}_m(B_\mathsf{X})^{1/m}\qquad\mathrm{yet}\qquad \mathrm{iq}(L)\lesssim\sqrt m.$$

Here $B_\mathsf{X}$ is the unit ball of the unitarily invariant normed space under consideration. The conjecture asserts that, after a volume-preserving linear position, every origin-symmetric convex body contains a comparably large body whose isoperimetric quotient is within a universal factor of the Euclidean optimum. The paper establishes the corresponding statement for unit balls of unitarily invariant matrix norms.