Recorded 09 February 2022. Emanuel Milman of Technion - Israel Institute of Technology presents "The log-Minkowski Problem" at IPAM's Calculus of Variations in Probability and Geometry Workshop. Abstract: The classical Minkowski problem asks to find a convex body K in Rn having a prescribed surface-area measure, which boils down to solving the Monge-Amp\`ere equation on the unit sphere Sn−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 Lp version for general p∈R (with the classical case corresponding to p=1) was suggested and publicized by E.~Lutwak. 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.