Related papers: The $Lp$ chord Minkowski problem for negative $p$

Let $K\subset R^n$ be a compact basic semi-algebraic set. We provide a necessary and sufficient condition (with no a priori bounding parameter) for a real sequence $y=(y_\alpha)$, $\alpha\in N^n$, to have a finite representing Borel measure…

Let $\mu$ be a nonnegative Borel measure on the open unit disk $\mathbb{D}\subset\mathbb{C}$. This note shows how to decide that the M\"obius invariant space $\mathcal{Q}_p$, covering $\mathcal{BMOA}$ and $\mathcal{B}$, is boundedly (resp.,…

Ben Andrews classified the limiting shape for isotropic curvature flow corresponding to the solutions of the $L_p$-Minkowski problem as $p\to-\infty$ in the planar case. In this paper, we use the group-invariant method to study the…

Three new combinations of convex bodies are introduced and studied: the $L_p$ fiber, $L_p$ chord and graph combinations. These combinations are defined in terms of the fibers and graphs of pairs of convex bodies, and each operation…

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…

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…

The Minkowski problem for torsional rigidity ($2$-torsional rigidity) was firstly studied by Colesanti and Fimiani \cite{CA} using variational method. Moreover, Hu \cite{HJ00} also studied this problem by the method of curvature flows and…

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, we solve the even capillary $L_p$-Minkowski problem for the range $-n < p < 1$ and $\theta \in (0,\frac{\pi}{2})$. Our approach is based on an iterative scheme that builds on the solution to the capillary Minkowski problem…

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,…

The uniqueness of the $L_p$-Minkowski problem has been a long standing problem in convex geometry. In the groundbreaking paper by Brendle-Choi-Daskalopoulos (Acta Math, {\bf219}, 2017), a full uniqueness result was shown for the subcritical…

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 prove a Pr\'ekopa-Leindler type inequality of the $L_p$ Brunn-Minkowski inequality. It extends an inequality proved by Das Gupta [8] and Klartag [16], and thus recovers the Pr\'ekopa-Leindler inequality. In addition, we…

In this paper, we prove a sharp anisotropic $L^p$ Minkowski inequality involving the total $L^p$ anisotropic mean curvature and the anisotropic $p$-capacity, for any bounded domains with smooth boundary in $\mathbb{R}^n$. As consequences,…

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…

In this paper, we study the $L_p$-Gaussian Minkowski problem, which arises in the $L_p$-Brunn-Minkowski theory in Gaussian probability space. We use Aleksandrov's variational method with Lagrange multipliers to prove the existence of the…

For $1 \leq p < \infty$, Ludwig, Haberl and Parapatits classified $L_p$ Minkowski valuations intertwining the special linear group with additional conditions such as homogeneity and continuity. In this paper,a complete classification of…

Boundedness of an abstract formulation of Hardy operators between Lebesgue spaces over general measure spaces is studied and, when the domain is L^1, shown to be equivalent to the existence of a Hardy inequality on the half line with…

Elementary proofs of sharp isoperimetric inequalities on a normed space $(\mathbb{R}^n,||\cdot||)$ equipped with a measure $\mu = w(x) dx$ so that $w^p$ is homogeneous are provided, along with a characterization of the corresponding…

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…