The Minkowski type problems for unbounded convex sets

The unbounded convex sets have important applications in differential geometry, commutative algebra, complex analysis and singularity theory, etc. Although there are some similarities between convex bodies and unbounded convex sets, recent results have shown strong differences between them as well. For example, Schneider established the Brunn-Minkowski inequality for unbounded convex C-close sets; however, the direction of this inequality is opposite to the classical Brunn-Minkowski inequality for convex bodies. Schneider also defined the surface area measure and hence posed the related Minkowski problem for unbounded convex C-close sets; but the solutions to this Minkowski problem look rather different from those to the classical Minkowski problem for convex bodies.

In this talk, I will discuss our recent progress on the geometric theory for unbounded convex sets. In particular, I will talk about the Minkowski type problems and present our solutions to these problems.