A sharp centro-affine isospectral inequality of Szegö–Weinberger type and the Lᵖ-Minkowski problem

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 C²-smooth origin-symmetric convex body K in Rⁿ. 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 Lᵖ-Minkowski problem in the subcritical range -n < p < 0, as well as new rigidity results for the critical exponent $p = -n$ and supercritical regime p < -n. In particular, we show that any K as above, which is not an ellipsoid, is a witness to non-uniqueness in the even Lᵖ-Minkowski problem for all p ∈ (-n, pK) and some pK ∈ (-n, 0), and that K can be chosen so that pK is arbitrarily close to $0$.