Centro-affine differential geometry and the log-Minkowski problem

  • Emanuel Milman

    Technion - Israel Institute of Technology, Haifa, Israel
Centro-affine differential geometry and the log-Minkowski problem cover

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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 is a centro-affine unit sphere, it has constant centro-affine Ricci curvature equal to , 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 - and log-Minkowski problems, as well as the corresponding global - and log-Minkowski conjectured inequalities. As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body in , there exists an origin-symmetric convex body with such that satisfies the log-Minkowski conjectured inequality, and such that is uniquely determined by its cone-volume measure . If is not extremely far from a Euclidean ball to begin with, an analogous isometric result, where is replaced by , is obtained as well.

Cite this article

Emanuel Milman, Centro-affine differential geometry and the log-Minkowski problem. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1386