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We establish a sharp upper-bound for the first non-zero even eigenvalue(corresponding to an even eigenfunction) of the Hilbert-Brunn-Minkowskioperator associated to a strongly convex $C^2$-smooth origin-symmetric convexbody $K$ in $\mathbb{R}^n$. Our isospectral inequality is centro-affineinvariant, attaining equality if and only if $K$ is a (centered) ellipsoid;this is reminiscent of the (non affine invariant) classicalSzeg\"{o}--Weinberger isospectral inequality for the Neumann Laplacian. The newupper-bound complements the conjectural lower-bound, which has been shown to beequivalent to the log-Brunn-Minkowski inequality and is intimately related tothe uniqueness question in the even log-Minkowski problem. As applications, weobtain new strong non-uniqueness results in the even $L^p$-Minkowski problem inthe subcritical range $-n < p < 0$, as well as new rigidity results for thecritical exponent $p=-n$ and supercritical regime $p < -n$. In particular, weshow that any $K$ as above which is not an ellipsoid is a witness tonon-uniqueness in the even $L^p$-Minkowski problem for all $p \in (-n,p_K)$ andsome $p_K \in (-n,0)$, and that $K$ can be chosen so that $p_K$ is arbitrarilyclose to $0$.

A sharp centro-affine isospectral inequality of Szeg\"{o}--Weinberger type and the $L^p$-Minkowski problem