Completness of Statistical Structures

Opozda, Barbara

doi:10.3390/math7010104

Cited by 1 publication

(1 citation statement)

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“…We refer to [85,12] for a detailed exposition and further information regarding affine differential geometry. For a development of the theory from the point of view of relative normalizations, we refer to [97,65,86], and from the point of view of statistical structures, we refer to [87,88].…”

Section: Affine Differential Geometrymentioning

confidence: 99%

Centro-Affine Differential Geometry and the Log-Minkowski Problem

Milman¹

2021

Preprint

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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 R n is a centro-affine unitsphere (having constant centro-affine Ricci curvature equal to n − 2), 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. This extends the results of Brendle-Choi-Daskalopoulos on the uniqueness of self-similar solutions to the power-of-Gauss-curvature flow from the isotropic to the pinched anisotropic case (for origin-symmetric solutions). As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body K in R n , there exists an origin-symmetric convex body K with K ⊂ K ⊂ 8 K, so that K satisfies the log-Minkowski conjectured inequality, and so that K is uniquely determined by its cone-volume measure V K . Analogous isometric results are derived as well.

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Section: Affine Differential Geometrymentioning

confidence: 99%