Smooth solutions to the $$L_p$$ L p dual Minkowski problem

On bm$(p,q)$-John ellipsoids

“…L p 表面积测度、 L p 积分曲率和对偶曲率测度是 (p, q) 阶对偶曲率测度 的特殊情形. 文献 [33][34][35][36] 研究了相应的 Minkowski 问题. Lutwak 等 [32] 利用 (p, q) 阶对偶曲率测度定 义了 L p 对偶混合体积, 统一了 L p 混合体积和对偶混合体积.…”

General (p,q)-mixed projection bodies

“…In the same way that the L p surface area and dual curvature measures, respectively, play a critical role in the L p and dual Brunn-Minkowski theories, the L p dual curvature measures can be seen to be a central concept within the unifying theory. See, e.g., [43][44][45][46] for some recent developments related to the L p dual curvature measures.…”

A flow method for a generalization of $ L_{p} $ Christofell-Minkowski problem

“…Dual curvature measures and their associated variational formulas was firstly introduced by Huang, Lutwak, Yang and Zhang in their recent groundbreaking work [8]. Then the dual curvature measures were extended into the L p case in [13] and then studied in [1,4,9].…”

“…On the other hand, anisotropic curvature flows provide alternative methods to prove the existences of elliptic PDEs arising from convex geometry, see e.g. [2,4,10,12,15].…”

“…The scalar function η(t) in (1.2) is usually used to keep M t normalized in a certain sense, see for examples [4,10,15]. In this paper, η is given by…”