Virtual Talk: The log-Minkowski Problem

Emanuel Milman
Technion - Israel Institute of Technology

The classical Minkowski problem asks to find a convex body $K$ in $\mathbb{R}^n$ having a prescribed surface-area measure, which boils down to solving the Monge-Amp\`ere equation on the unit sphere $S^{n-1}$. Existence and regularity were extensively studied by Minkowski, Alexandrov, Lewy, Nirenberg, Cheng--Yau, Pogorelov, Caffarelli and many others; uniqueness (up to translation) is an immediate consequence of the classical Brunn--Minkowski inequality. An analogous $L^p$ version for general $p \in \mathbb{R}$ (with the classical case corresponding to $p=1$) was suggested and publicized by E.~Lutwak.

The $L^p$-Minkowski problem is related to numerous other fields:
\item Non-linear PDE: it is a Monge-Amp\`ere--type equation on $S^{n-1}$.
\item Geometric Flows: it describes self-similar solutions to the anisotropic $\alpha$-power-of-Gauss-curvature flow (for $\alpha = \frac{1}{1-p}$).
\item Calculus of Variations: it is the Euler-Lagrange equation for the $L^p$-Minkowski functional.
\item Brunn--Minkowski theory: it is related to a strengthening of the classical Brunn--Minkowski inequality.

The case $p \geq 1$ is well understood, but the case $p < 1$ is much more challenging due to a lack of a corresponding $L^p$-Brunn--Minkowski theory. In particular, no uniqueness is possible in general, but it was conjectured by B\"or\"oczky--Lutwak--Yang--Zhang that for \emph{origin-symmetric} convex bodies $\mathcal{K}_e$, uniqueness in the $L^p$-Minkowski problem should hold for all $p \in [0,1)$. Equivalently, the $L^0-$ (log-)Minkowski functional should have a unique global minimum on $\mathcal{K}_e$, the log-Brunn-Minkowski inequality should hold on $\mathcal{K}_e$, and the anisotropic Gauss-curvature flow should have a unique origin-symmetric self-similar solution.

We report on recent progress towards this conjecture. In particular, we resolve the isomorphic version of the log-Minkowski problem, and extend 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). Our main new tool is an interpretation of the problem as a spectral question in centro-affine differential geometry.

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