Continuity of the solution to the even logarithmic Minkowski problem in the plane

Abstract

In this paper, it is proved that the weak convergence of the L p Guassian surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric for p ≥ 1. Moreover, this paper obtains the solution to the L p Guassian Minkowski problem is continuous with respect to p.2000 Mathematics Subject Classification. 52A40. Key words and phrases. convex body; continuity; L p Gaussian surface area measure; L p Gaussian Minkowski problem.

The affine log-Aleksandrov–Fenchel inequality

“…Recently, the log-Minkowski inequality and the log-Aleksandrov-Fenchel inequality and its dual form have attracted extensive attention and research: see references [1,3,4,6,7,8,12,13,14,18,17,19,20,21,22,23]. In this paper, we generalize the log-Minkowski inequality (1.1) and the log-Aleksandrov-Fenchel inequality (1.2) to the mixed affine quermassintegrals.…”

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

“…Huang 等 [26] 引入了对偶曲率测 度. 对偶曲率测度受到密切关注, 相关进展可参见文献 [27][28][29][30][31]. 最近, Lutwak 等 [32] 定义了 L p 对偶曲 率测度 ((p, q) 阶对偶曲率测度).…”

Lp$L_p$‐cosine transform of the (p,q)${(p,q)}$‐th dual curvature measure

“…Moreover, the dual curvature measure defined in [11] is a special case of the $$L_p$$ dual curvature measures. For the Minkowski problems for $$L_p$$ dual curvature measures, one can see [1, 3, 8, 14, 30, 31, 35, 36, 41]. $$(p,q)$$‐mixed geominimal surface area and $$(p,q)$$‐mixed affine surface area are introduced and associate isoperimetric inequalities are obtained in [15].…”