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- W4322825304 abstract "We interpret the log-Brunn-Minkowski conjecture of Boroczky-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 $mathbb{R}^n$ 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 $bar K$ in $mathbb{R}^n$, there exists an origin-symmetric convex body $K$ with $bar K subset K subset 8 bar K$, so that $K$ satisfies the log-Minkowski conjectured inequality, and so that $K$ is uniquely determined by its cone-volume measure $V_K$. If $bar K$ is not extremely far from a Euclidean ball to begin with, an analogous isometric result, where $8$ is replaced by $1+epsilon$, is obtained as well." @default.
- W4322825304 created "2023-03-03" @default.
- W4322825304 creator A5052187675 @default.
- W4322825304 date "2021-04-26" @default.
- W4322825304 modified "2023-09-27" @default.
- W4322825304 title "Centro-Affine Differential Geometry and the Log-Minkowski Problem" @default.
- W4322825304 doi "https://doi.org/10.48550/arxiv.2104.12408" @default.
- W4322825304 hasPublicationYear "2021" @default.
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