The cone-volume functional was originally introduced by Lutwak, Yang and Zhang (LYZ) in 2001 to attack the celebrated longstanding Schneider projection problem in convex geometry. It is closely related to the cone-volume measure of convex bodies and has strong applications to the reverse affine isoperimetric problem and the logarithmic Minkowski problem.
In this talk, we will first review the fundamental properties of the cone-volume functional and its applications to the Schneider projection problem. Then we will talk on the solved LYZ conjecture for the cone-volume functional and its applications to the logarithmic Minkowski problem. Finally, we will report our very recent results on its generalizations, including the variational formula and the extreme problem on the mixed cone-volume functional.
This talk is based on the joint work with Qiang Sun and Ge Xiong.
鲁新宝，复旦大学上海数学中心博士后，博士毕业于同济大学，研究方向为凸几何与几何分析，目前已发表学术论文3篇，包括Isreal J. Math., Indiana Univ. Math. J.等高水平期刊。