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 L_p addition of convex bodies, Lutwak derived the L_p surface area measure and initiated the study of the L_p Minkowski problem in 1993. Note that solving the L_p Minkowski problem requires to find (weak) solutions to some Monge-Ampere equations.

In this talk, we will discuss the L_p 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 L_p addition of C-coconvex sets, the L_p Brunn-Minkowski and Minkowski inequalities, and a variational formula which derives the L_p 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.

Speaker: 

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.

Zoom:

Link: https://us02web.zoom.us/j/81477927989?pwd=WVJ2Ri91bzNjM3FyRGlKMnJpOE9aQT09