A sharp centro-affine isospectral inequality of Szegö–Weinberger type and the $L^p$-Minkowski problem

Abstract

We establish a sharp upper-bound for the first non-zero eigenvalue corresponding to an even eigenfunction of the Hilbert–Brunn–Minkowski operator (the centro-affine Laplacian) associated to a strongly convex $C^2$-smooth origin-symmetric convex body $K$ in $\mathbb{R}^n$. Our isospectral inequality is centro-affine invariant, attaining equality if and only if $K$ is a (centered) ellipsoid; this is reminiscent of the (non affine invariant) classical Szegö–Weinberger isospectral inequality for the Neumann Laplacian. The new upperbound complements the conjectural lower-bound, which has been shown to be equivalent to the $\log$-Brunn–Minkowski inequality and is intimately related to the uniqueness question in the even $\log$-Minkowski problem. As applications, we obtain new strong non-uniqueness results for the even $L^p$-Minkowski problem in the subcritical range $-n \lt p \lt 0$, as well as new rigidity results for the critical exponent $p = -n$ and supercritical regime $p \lt -n$. In particular, we show that any $K$ as above, which is not an ellipsoid, is a witness to non-uniqueness in the even $L^p$-Minkowski problem for all $p \in (-n, p_K)$ and some $p_K \in (-n, 0)$, and that $K$ can be chosen so that $p_K$ is arbitrarily close to $0$.