The classical Minkowski problem was posed by Minkowski in 1897 and has great influence on convex geometry and partial differential equations. It asks whether a given nonzero finite Borel measure on the unit sphere can be the surface area measure of some convex body. Based on the Lp addition of convex bodies, Lutwak derived the Lp surface area measure and initiated the study of the Lp Minkowski problem in 1993. Note that solving the Lp Minkowski problem requires to find (weak) solutions to some Monge-Ampere equations.
In this talk, we will discuss the Lp Brunn-Minkowski theory for C-coconvex sets. The C-coconvex sets have found important applications in many fields, such as algebraic geometry, singularity theory, etc. We will talk about the Lp addition of C-coconvex sets, the Lp Brunn-Minkowski and Minkowski inequalities, and a variational formula which derives the Lp surface area measure for C-coconvex sets. The related Minkowski problem will be presented and its solution will be provided.
We also discuss the case for p=0 which leads to the log-Minkowski problem for C-coconvex sets. In particular, we will discuss the log-Brunn-Minkowski and log-Minkowski inequalities for C-coconvex sets. These inequalities solve an open problem regarding the uniqueness of the solutions to the log-Minkowski problem for C-coconvex sets raised by Schneider.
Dr. Deping Ye is a full professor at Memorial University, Canada. He received his PhD degree from Case Western Reserve University, USA. His research interests are convex geometry, quantum information and their applications.
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