The dual curvature measures for convex bodies were first introduced by Huang-Lutwak-Yang -Zhang in 2016, and they derived the Aleksandrov variational formula for dual quermassintergrals. With the help of the variational formula, they solved the associated Minkowski problem.
It is well-known that the set of convex bodies can be embedded in the set of log-concave functions via their characteristic functions. So the first speaker Jiaqian Liu(a joint work with Huang, Xi and Zhao) generalized dual curvature measures for log-concave functions and use Aleksandrov variational method to obtain the existence of the solution to the functional dual Minkowski problem.
In 2024, Feng-Li-Liu introduced the dual Minkowski problem in Gaussian space. They solved the associated Minkowski problem in even cases by utilizing varational method and continuity method (or degree theory) respectively. In this talk, the second speaker Shengyu Tang solved this problem by variational method without o-symmetric condition, and degree theory for a broader range. This also constructed two distinct solutions, which implies the non-uniqueness of the dual Minkowski problem in Gaussian space.
