A sharp centro-affine isospectral inequality of Szegö-Weinberger type and the \(L^p\)-Minkowski problem (Q6562509)

The classical Minkowski problem asks about the existence and uniqueness of a convex body with prescribed surface-area measure modulo natural conditions. Its positive solution is a classical result of Brunn-Minkowski theory. It is the case \(p=1\) of the Minkowski problem for the \(L^p\) surface-area measure which is more challenging and is related with the spectral analysis of the Hilbert-Brunn-Minkowski operator.\N\NThe author establishes a sharp upper-bound for the first non-zero eigenvalue of the operator corresponding to an even eigenfunction. The new upper-bound 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, some new strong non-uniqueness results for the even \(L^p\)-Minkowski problem are obtained. In particular, it is shown that any body which is not an ellipsoid, is a witness to non-uniqueness in the even \(L^p\)-Minkowski problem (for some specific values of \(p\)).