Abstract
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–Ampère 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:
- **Non-linear PDE:** it is a Monge–Ampère–type equation on \( S^{n-1} \).
- **Geometric flows:** it describes self-similar solutions to the anisotropic \( \alpha \)-power-of-Gauss-curvature flow (for \( \alpha = \frac{1}{1-p} \)).
- **Calculus of variations:** it is the Euler–Lagrange equation for the \( L^p \)-Minkowski functional.
- **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 Böröczky–Lutwak–Yang–Zhang that for *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.
The \( L^p \)-Minkowski problem is related to numerous other fields:
- **Non-linear PDE:** it is a Monge–Ampère–type equation on \( S^{n-1} \).
- **Geometric flows:** it describes self-similar solutions to the anisotropic \( \alpha \)-power-of-Gauss-curvature flow (for \( \alpha = \frac{1}{1-p} \)).
- **Calculus of variations:** it is the Euler–Lagrange equation for the \( L^p \)-Minkowski functional.
- **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 Böröczky–Lutwak–Yang–Zhang that for *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.