Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Centro-Affine Differential Geometry and the Log-Minkowski Problem   We interpret the log-Brunn-Minkowski conjecture of
B\"or\"oczky-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, analogous isometric results
are derived as well.

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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat