Centro-affine differential geometry and the log-Minkowski problem

We interpret the log-Brunn–Minkowski conjecture of Böröczky–Lutwak–Yang–Zhang as a spectral problem in centro-affine differential geometry. In particular, we show that the Hilbert–Brunn–Minkowski operator coincides with the centro-affine Laplacian, thus obtaining a new avenue for tackling the conjecture using insights from affine differential geometry. As every strongly convex hypersurface in Rⁿ is a centro-affine unit sphere, it has constant centro-affine Ricci curvature equal to n - 2, in stark contrast to the standard weighted Ricci curvature of the associated metric-measure space, which will in general be negative. In particular, we may use the classical argument of Lichnerowicz and a centro-affine Bochner formula to give a new proof of the Brunn–Minkowski inequality. For origin-symmetric convex bodies enjoying fairly generous curvature pinching bounds (improving with dimension), we are able to show global uniqueness in the L^p- and log-Minkowski problems, as well as the corresponding global L^p- and log-Minkowski conjectured inequalities. As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body K̄ in Rⁿ, there exists an origin-symmetric convex body K with K̄ ⊂ K ⊂ 8K̄ such that K satisfies the log-Minkowski conjectured inequality, and such that K is uniquely determined by its cone-volume measure V_K. If K̄ is not extremely far from a Euclidean ball to begin with, an analogous isometric result, where 8 is replaced by 1 C ", is obtained as well.