I primarily work in the field of convex geometric analysis. I am particularly interested in the Minkowski-type problems for geometric measures obtained by “differentiating” geometric invariants. These problems generate Monge-Ampère equations on the unit sphere and can be tackled from a PDE point of view (e.g., geometric flow).
I am also interested in isoperimetric inequalities (both affine and non-affine, geometric and functional).
- (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, in press.
- (with K. Böröczky, E. Lutwak, D. Yang, and G. Zhang) The dual Minkowski problem for symmetric convex bodies. Adv. Math. 356: 106805, 2019.
- (with C. Chen and Y. Huang) Smooth solutions to the $L_p$ dual Minkowski problem. Math. Ann. 373 (3-4):953-976, 2019.
- The $L_p$ Aleksandrov problem for origin-symmetric polytopes. Proc. Amer. Math. Soc. 147 (10):4477-4492, 2019.
- (with Y. Huang) On the $L_p$ dual Minkowski problem. Adv. Math. 332: 57-84, 2018.
- Existence of solutions to the even dual Minkowski problem. J. Differential Geom. 110 (3): 543–572, 2018.
- The dual Minkowski problem for negative indices. Calc. Var. Partial Differential Equations 56:18,2017. arXiv
- On $L_p$-affine surface area and curvature measures. Int. Math. Res. Not. IMRN (5): 1387-1423, 2016. arXiv