相关论文: The Dual Minkowski Problem under Group Actions
In this paper, we study the $L_p$ dual Minkowski problem for all $q, p \in \mathbb{R}$ from an algebraic perspective. We establish the existence of solutions for group-invariant convex bodies (not necessarily origin-symmetric), thereby…
This paper studies the general Lp dual curvature density equation under a group symmetry assumption. This geometric partial differential equation arises from the general Lp dual Minkowski problem of prescribing the Lp dual curvature measure…
The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in…
We prove tight subspace concentration inequalities for the dual curvature measures $\widetilde{\mathrm{C}}_q(K,\cdot)$ of an $n$-dimensional origin-symmetric convex body for $q\geq n+1$. This supplements former results obtained in the range…
The central focus of this paper is the $L_p$ dual Minkowski problem for $C$-compatible sets, where $C$ is a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior. Such a problem deals with the characterization of the $(p,…
Given a real number $q$ and a star body in the $n$-dimensional Euclidean space, the generalized dual curvature measure of a convex body was introduced by Lutwak-Yang-Zhang [43]. The corresponding generalized dual Minkowski problem is…
In this paper, combining the covolume, we study the Minkowski theory for the non-compact convex set with an asymptotic boundary condition. In particular, the mixed covolume of two non-compact convex sets is introduced and its geometric…
In this paper, the $q$-th dual curvature measure is extended to convex functions and the associated Minkowski problem is posed. A special case includes the $q$-th dual curvature measure of convex bodies which defined by Huang, Lutwak, Yang…
This paper studies the core problems in the $L_p$ dual Brunn-Minkowski theory, encompassing the $L_p$ Minkowski problem and $L_p$ Brunn-Minkowski inequality for dual quermassintegrals. For the case $0<p<q\leq n$, we establish $C^0$…
In Convex Geometry, a core topic is the $L_p$-Minkowski problem \begin{equation}\label{e0.1} \det(\nabla^2h+hI)=fh^{p-1}, \ \ \forall X\in{\mathbb{S}}^n, \ \ \forall p\in \mathbb{R} \end{equation} of Monge-Amp\`{e}re type. By the…
We prove the log-Brunn-Minkowski conjecture for convex bodies with symmetries to $n$ independent hyperplanes, and discuss the equality case and the uniqueness of the solution of the related case of the logarithmic Minkowski problem. We also…
Recently, the duals of Federer's curvature measures, called dual curvature measures, were discovered by Huang, Lutwak, Yang, and Zhang (ACTA, 2016). In the same paper, they posed the dual Minkowski problem, the characterization problem for…
Existence of symmetric (resp. asymmetric) solutions to the $L_p$ Gaussian Minkowski problem for $p\leq 0$ (resp. $p\geq 1$) will be provided. Moreover, existence and uniqueness of smooth solutions to the problem for $p>n$ will also be…
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…
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…
The dual $L_p$-Minkowski problem with $p<0<q$ is investigated in this paper. By proving a new existence result of solutions and constructing an example, we obtain the non-uniqueness of solutions to this problem.
The classical Minkowski problem in Minkowski space asks, for a positive function $\phi$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $\phi(\eta)^{-1}$ is the Gauss--Kronecker…
For two positive integers m and n, we let ${\mathcal P}_n$ be the open convex cone in ${\mathbb R}^{n(n+1)/2}$ consisting of positive definite n x n real symmetric matrices and let ${\mathbb R}^{(m,n)}$ be the set of all m x n real…
The classical Minkowski problem for convex bodies has deeply influenced the development of differential geometry. During the past several decades, abundant mathematical theories have been developed for studying the solutions of the…
We give a systematic and thorough study of geometric notions and results connected to Minkowski's measure of symmetry and the extension of the well-known Minkowski functional to arbitrary, not necessarily symmetric convex bodies K on any…