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-2002778 ($159,159.00, 06/2020—05/2023).
- (with Y. Huang and D. Xi) The Minkowski problem in Gaussian probability space. submitted. pdf
- (with D. Xi) General affine invariances related to Mahler volume. submitted.
- (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