Related papers: The discrete logarithmic Minkowski problem for the…
The Minkowski problem in Gaussian probability space is studied in this paper. In addition to providing an existence result on a Gaussian-volume-normalized version of this problem, the main goal of the current work is to provide uniqueness…
General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures…
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…
We extend Newton's problem of minimal resistance to the Lorentz-Minkowski space. We derive the functional energy and determine the Euler-Lagrange equation. In contrast to the Euclidean case, this equation is quasilinear elliptic, and thus,…
We provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new…
In this paper, we establish a necessary condition for the logarithmic Minkowski problem in higher dimensions. This result generalizes a necessary condition proposed by Liu, Lu, Sun, and Xiong in their investigation of the two-dimensional…
We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of the $n$-dimensional Euclidean space. For the existence part we…
Minkowski sums are of theoretical interest and have applications in fields related to industrial backgrounds. In this paper we focus on the specific case of summing polytopes as we want to solve the tolerance analysis problem described in…
Let $K$ be a smooth, origin-symmetric, strictly convex body in $\mathbb{R}^n$. If for some $\ell\in GL(n,\mathbb{R})$, the anisotropic Riemannian metric $\frac{1}{2}D^2 \Vert\cdot\Vert_{\ell K}^2$, encapsulating the curvature of $\ell K$,…
The existence of solutions to the Gaussian logarithmic Minkowski problem for C-pseudo-cones is established in this paper. In addition, the non-uniqueness of solutions to the problem is demonstrated.
In this paper, we study the dual Minkowski problem under group symmetry. For $0<q\le n$, we give a complete existence characterization in the framework of $G$-invariant convex bodies when the group $G\subset O(n)$ has no nonzero fixed…
In the case of symmetries with respect to n independent linear hyperplanes, a stability version of the logarithmic Brunn-Minkowski inequality and the logarithmic Minkowski inequality for convex bodies is established.
The Dirichlet boundary value problem for the Stokes operator with $L^p$ data in any dimension on domains with conical singularity (not necessary a Lipschitz graph) is considered. We establish the solvability of the problem for all $p\in…
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…
We study the Minkowski problem corresponding to the p-harmonic measures and obtain results previously known for harmonic measures due to Jerison. We show that a class of Borel measures on spheres can be prescribed by p-harmonic measures on…
By studying $L^p$-combinations of strongly isomorphic polytopes, we prove the equivalence of the $L^p$-Brunn-Minkowski inequality conjectured by B\"or\"oczky, Lutwak, Yang and Zhang to the local version of the inequality studied by…
We consider a system of PDEs of Monge-Kantorovich type arising from models in granular matter theory and in electrodynamics of hard superconductors. The existence of a solution of such system (in a regular open domain…
The horospherical $p$-Christoffel-Minkowski problem was posed by Li and Xu (2022) as a problem prescribing the $k$-th horospherical $p$-surface area measure of $h$-convex domains in hyperbolic space $\mathbb{H}^{n+1}$. It is a natural…
The multistsochastic Monge--Kantorovich problem on the product $X = \prod_{i=1}^n X_i$ of $n$ spaces is a generalization of the multimarginal Monge--Kantorovich problem. For a given integer number $1 \le k<n$ we consider the minimization…
The study of the dual curvature measures [Y. Huang, E. Lutwak, D. Yang \& G. Y. Zhang, Acta. Math. 216 (2016): 325-388], which connects the cone-volume measure and Aleksandrov's integral curvature, and has created a precedent for the…