The Logarithmic Minkowski conjecture and the $L_p$-Minkowski Problem
Colloque / Congrès / Forum
Ouvert au grand public
The Minkowski problem forms the core of various areas in fully nonlinear partial differential equations on the sphere and convex geometry, and is intimately connected to the Brunn-Minkowski inequality about convex bodies. Recently Lutwak's $L_p$-Minkowski problem and its variants have received significant attention where the $p=1$ case corresponds to the classical Minkowski problem, and the case $p=0$ is the so-called Logarithmic Minkowski problem going back to Firey. The talk surveys developments about the $L_p$-Minkowski problem and the central Logarithmic Minkowski Conjecture concerning the uniqueness of the even solution in the $p=0$ case, and the relation of this conjecture to some strengthening of the Brunn-Minkowski inequality for origin symmetric convex bodies, to some inequalities for the Gaussian density, and to some spectral gap estimates for certain self-adjoint operators.