In this paper we study the the Gauss image problem, which is a generalization of the Aleksandrov problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain the existence of smooth solutions to this problem.

Smooth solutions to the Gauss image problem

In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure $\widetilde{C}_{G,\Psi,\lambda}(\Omega,\cdot)$ of convex body $\Omega$ in $\mathbb{R}^{n+1}$ containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-studied curvature flows related to Minkowski type problems as special cases. Our results not only generalize many previously known solutions to the Minkowski type and Gauss image problems, but also provide solutions to those problems in many unsolved cases.

A flow approach to the Musielak-Orlicz-Gauss image problem

. The dual Minkowski problem in the two-dimensional plane is studied in this paper. By combining the theoretical analysis and numerical estimation of an integral with parameters, we ﬁnd the number of solutions to this problem for the constant dual curvature case when 0 < q ≤ 4. An improved nonuniqueness result when q > 4 is also obtained. As an application, a result on the uniqueness and nonuniqueness of solutions to the L p -Alexandrov problem is obtained for p < 0.

On the number of solutions to the planar dual Minkowski problem

Existence of symmetric solutions to the Gaussian Minkowski problem was established by Huang et al. In this paper, we show the existence of non-symmetric solutions to this problem by studying the related Monge–Ampère type equation on the sphere. 

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Existence of Non-symmetric Solutions to the Gaussian Minkowski Problem

In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of convex bodies. For a convex body $K$, its Musielak-Orlicz-Gauss image measure, denoted by $\widetilde{C}_{\Theta}(K, \cdot)$, involves a triple $\Theta=(G, \Psi, \lambda)$ where $G$ and $\Psi$ are two Musielak-Orlicz functions defined on $S^{n-1}\times (0, \infty)$ and $\lambda$ is a nonzero finite Lebesgue measure on the unit sphere $S^{n-1}$. Such a measure can be produced by a variational formula of $\widetilde{V}_{G, \lambda}(K)$ (the general dual volume of $K$ with respect to $\lambda$) under the perturbations of $K$ by the Musielak-Orlicz addition defined via the function $\Psi$. The Musielak-Orlicz-Gauss image problem contains many intensively studied Minkowski type problems and the recent Gauss image problem as its special cases. Under the condition that $G$ is decreasing on its second variable, the existence of solutions to this problem is established.

On the Musielak-Orlicz-Gauss image problem

In this paper we study the dual Orlicz-Minkowski problem, which is a generalization of the dual Minkowski problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain a new existence result of smooth even solutions to this problem for smooth even measures.

Existence of smooth even solutions to the dual Orlicz-Minkowski problem

In this paper we study a normalized anisotropic Gauss curvature flow of strictly convex, smooth closed hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Ampere type equation and we establish a new existence result of solutions to the dual Orlicz-Minkowski problem for even smooth measures.

Ni Xiang

Papers

Existence of smooth even solutions to the dual Orlicz-Minkowski problem

Smooth solutions to the even dual Orlicz-Minkowski problem