Related papers: The $L_{p}$ Gaussian Minkowski problem for $C$-pse…

We study several of the recent conjectures in regards to the role of symmetry in the inequalities of Brunn-Minkowski type, such as the $L_p$-Brunn-Minkowski conjecture of B\"or\"oczky, Lutwak, Yang and Zhang, and the Dimensional…

Necessary and sufficient conditions are given for the existence of solutions to the discrete Lp Minkowski problem for the critical case where 0 < p < 1.

Existence and uniqueness of the solution to the discrete Lp Minkowski problem for $\mathfrak{p}$-capacity are proved when $p \geq 1$ and $1<\mathfrak{p}<n$. For general Lp Minkowski problem for $\mathfrak{p}$-capacity, existence and…

In this paper, we investigate an $L_p$ dual Christoffel-Minkowski type problem for the Hessian quotient operator $\frac{\sigma_{k}(\Lambda)}{\sigma_{l}(\Lambda)}$, where the operator $\Lambda$ has been widely studied in the literature.…

In this paper, the $L_q$-Minkowski problem of anisotropic $p$-torsional rigidity is considered. The existence of the solution of the $L_q$-Minkowski problem of anisotropic $p$-torsional rigidity with $0<q<1$ and $1<q\neq \frac{p}{p-1}+n$ is…

This paper introduces the \textit{anisotropic $\omega_0$-capillary $p$-sum} of two hypersurfaces in $\mathbb{R}_+^{n+1}$, and establishes a theory for anisotropic capillary convex bodies. For a smooth convex hypersurface $\Sigma $ with…

An Orlicz version of the $L_p$-Minkowski problem on $S^{n-1}$ is discussed corresponding to the case $-n<p<0$.

We consider the porous medium equation on manifolds with conical singularities and show existence, uniqueness and maximal $L^{p}$-regularity of a short time solution. In particular, we obtain information on the short time asymptotics of the…

In contemporary convex geometry, the rapidly developing L_p-Brunn Minkowski theory is a modern analogue of the classical Brunn Minkowski theory. A cornerstone of this theory is the L_p-affine surface area for convex bodies. Here, we…

We consider a fully nonlinear partial differential equation associated to the intermediate $L^p$ Christoffel-Minkowski problem in the case $1<p<k+1$. We establish the existence of convex body with prescribed $k$-th even $p$-area measure on…

For $p\in (-\infty,0)\cup(0,1)$ and a convex body $K\subset\mathbb{R}^n$ with the origin in its interior, we construct the family of $p$-affine dual curvature measures $\mathcal{I}_p(K,\cdot)$ with respect to $K$. The affine-invariant…

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 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…

Recently, Huang and Qin \cite{HY01} introduced the Gaussian chord measure and $L_p$-Gaussian chord measure by variational methods. Meanwhile, they posed Gaussian chord Minkowski problem for $p=1$ and used variational methods to obtain an…

The $L_p$ chord Minkowski problem was recently introduced by Lutwak, Xi, Yang and Zhang, which seeks to determine the necessary and sufficient conditions for a given finite Borel measure such that it is the $L_p$ chord measure of a convex…

This paper describes the theory of Minkowski problems for geometric measures in convex geometric analysis. The theory goes back to Minkowski and Aleksandrov and has been developed extensively in recent years. The paper surveys classical and…

In this paper, we study the planar Lp-Minkowski problem for all p, which was introduced by Lutwak [23]. A detailed exploration on solvability and uniqueness will be presented.

$L_p$ Brunn-Minkowski type inequa\-li\-ties for the lattice point enumerator $\mathrm{G}_n(\cdot)$ are shown, both in a geometrical and in a functional setting. In particular, we prove that \[\mathrm{G}_n\bigl((1-\lambda)\cdot K +_p…

The Minkowski problem for a class of unbounded closed convex sets is considered. This is equivalent to a Monge-Amp\`ere equation on a bounded convex open domain with possibly non-integrable given data. A complete solution (necessary and…

In this article we carry out a detailed investigation of the geometric nature of the points at infinity of Minkowski superspace. It turns out that there are several sets of points forming the superconformal boundary of Minkowski superspace:…