I work in convex geometry. I am interested in the existence and uniqueness of solutions to a variety of Monge-Ampère type equations that come from the study of convex bodies. Solutions to these PDEs lead to the reconstruction of convex bodies when different types of geometric measures are prescribed. In convex geometry, they are known as Minkowski-type problems.
The techniques I use include calculus of variation, geometric flow, method of continuity, and degree theory.
I am also interested in sharp isoperimetric inequalities (both affine and non-affine, geometric and functional).
I am partially supported by a standard NSF grant: DMS-2132330 (transferred from DMS-2002778) 06/2020—05/2024.
- (with N. Fang, D. Ye and Z. Zhang) The dual Orlicz curvature measures for log-concave functions and their related Minkowski problems. submitted. arXiv
- (with Y. Huang, J. Liu and D. Xi) Dual curvature measures for log-concave functions. J. Differential Geom., accepted, 2023. arXiv
- (with S. Chen, S. Hu and W. Liu) On the planar Gaussian-Minkowski problem. Adv. Math., Volume 435, Part A: 109351, 2023. link
- (with L. Guo and D. Xi) The $L_p$ chord Minkowski problem in a critical interval. Math. Ann., 2023. link
- (with D. Xi, D. Yang and G. Zhang) The $L_p$ chord Minkowski problem. Advanced Nonlinear Studies, 23 (1): Paper No. 20220041, 22, 2023. link
- (with D. Xi) General affine invariances related to Mahler volume. International Mathematics Research Notices. IMRN, no. 18: 14151–14180, 2022. link
- (with Y. Huang and D. Xi) The Minkowski problem in Gaussian probability space. Advances in Mathematics, 385: Paper No. 107769, 36, 2021 pdf
- (with K. Böröczky, E. Lutwak, D. Yang, and G. Zhang) The Gauss image problem. Communications on Pure and Applied Mathematics, 73: 1406-1452, 2020. pdf
- (with K. Böröczky, E. Lutwak, D. Yang, and G. Zhang) The dual Minkowski problem for symmetric convex bodies. Adv. Math. 356: 106805, 2019. pdf
- (with C. Chen and Y. Huang) Smooth solutions to the $L_p$ dual Minkowski problem. Math. Ann. 373 (3-4):953-976, 2019. pdf
- The $L_p$ Aleksandrov problem for origin-symmetric polytopes. Proc. Amer. Math. Soc. 147 (10):4477-4492, 2019. pdf
- (with Y. Huang) On the $L_p$ dual Minkowski problem. Adv. Math. 332: 57-84, 2018. pdf
- Existence of solutions to the even dual Minkowski problem. J. Differential Geom. 110 (3): 543–572, 2018. pdf
- The dual Minkowski problem for negative indices. Calc. Var. Partial Differential Equations 56:18,2017. pdf arXiv
- On $L_p$-affine surface area and curvature measures. Int. Math. Res. Not. IMRN (5): 1387-1423, 2016. arXiv