# Centro-affine differential geometry and the log-Minkowski problem

### Emanuel Milman

Technion - Israel Institute of Technology, Haifa, Israel

## Abstract

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 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 $Kˉ$ in $R_{n}$, there exists an origin-symmetric convex body $K$ with $Kˉ⊂K⊂8Kˉ$ such that $K$ satisfies the log-Minkowski conjectured inequality, and such that $K$ is uniquely determined by its cone-volume measure $V_{K}$. If $Kˉ$ is not extremely far from a Euclidean ball to begin with, an analogous *isometric* result, where $8$ is replaced by $1+ε$, 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