Related papers: The Lp Minkowski problem for q-torsional rigidity
In this paper the dual Orlicz-Minkowski problem, a generalization of the $L_p$ dual Minkowski problem, is studied. By studying a flow involving the Gauss curvature and support function, we obtain a new existence result of solutions to this…
The second domain variation of the $p$-capacity and the $q$ - torsional rigidity for compact sets in $R^d, d\geq3$ with $1<p<d$ is computed. Conditions on $p$ and $q>1$ are given such that the ball is a local minimzer or maximizer of the…
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…
The mixed Christoffel-Minkowski problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, one of which, appearing multiple times, is free and…
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…
We consider the problem of lower bounding a generalized Minkowski measure of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp…
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.
In \cite{LX}, the first author and the third author introduced and studied the horospherical $p$-Minkowski problem for smooth horospherically convex domains in hyperbolic space. In this paper, we introduce and solve the discrete…
On the class of log-concave functions on $\R^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of…
Generalization of Lyapunov convexity theorem is proved for vector measure with values in Banach spaces with unconditional bases, which are q-concave for some $q<\infty.$
An Orlicz version of the $L_p$-Minkowski problem on $S^{n-1}$ is discussed corresponding to the case $-n<p<0$.
We study, in dimension $n\geq2$, the eigenvalue problem and the torsional rigidity for the $p$-Laplacian on convex sets with holes, with external Robin boundary conditions and internal Neumann boundary conditions. We prove that the annulus…
We consider the sharp Sobolev-Poincar\'e constant for the embedding of $W^{1,2}_0(\Omega)$ into $L^q(\Omega)$. We show that such a constant exhibits an unexpected dual variational formulation, in the range $1<q<2$. Namely, this can be…
For any bounded convex domain \Omega in R^N, we assign a positive finite Borel measure associated with the solution to a su-blinear elliptic equation in \Omega. We prove that this measure is weakly continuous in the sense of measure with…
Using an optimal containment approach, we quantify the asymmetry of convex bodies in $\mathbb{R}^n$ with respect to reflections across affine subspaces of a given dimension. We prove general inequalities relating these ''Minkowski…
In this paper we discuss some results regarding the rigidity of the Borell-Brascamp-Lieb inequality and the Brunn-Minkowski inequality. We show a theorem of rigidity on curvature and measure of the Borell-Brascamp-Lieb inequality, a…
In this paper we consider cases of existence of invariant measure, additional first integrals, and Poisson structure in a problem of rigid body's rolling without sliding on plane and sphere. The problem of rigid body's motion on plane was…
We study the iterations of a class of curvature image operators $\Lambda_p^{\varphi}$ introduced by the author in (J. Funct. Anal. 271 (2016) 2133--2165). The fixed points of these operators are the solutions of the $L_p$ Minkowski problems…
\footnotesize B\"{o}r\"{o}czky, Lutwak, Yang and Zhang recently conjectured a certain strengthening of the Brunn-Minkowski inequality for symmetric convex bodies, the so-called log-Brunn-Minkowski inequality. We establish this inequality…
The notion of $s$--fractional $L^p$ polar projection bodies, recently introduced by Haddad and Ludwig (Math.\ Ann.\ \textbf{388}:1091--1115, 2024), provides a bridge between fractional Sobolev theory and convex geometry. In this manuscript,…