Related papers: Smoothness in the $l_p$ Minkowski problem for $p<1…

We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants,…

The uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem in $\mathbb{R}^{n+1}$ is established when $-(n+1)<p<-1$ with $n\geq 1$, without requiring the origin-centred assumption on convex bodies.

Let $u\in W^{2,p}_0$, $1\le p\le \infty$ be a solution of the Poisson equation $\Delta u = h$, $h\in L^p$, in the unit disk. It is proved that $\|\nabla u\|_{L^p} \le a_p\|h\|_{L^p}$ with sharp constant $a_p$ for $p=1$ and $p=\infty$ and…

In this paper we study the dual Orlicz-Minkowski problem, which is a generalization of the dual Minkowski problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial…

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…

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,…

A previous study analyzed the convergence of probability densities for forward and inverse problems when a sequence of approximate maps between model inputs and outputs converges in $L^\infty$. This work generalizes the analysis to cases…

We consider a different $L^p$-Minkowski combination of compact sets in $\mathbb{R}^n$ than the one introduced by Firey and we prove an $L^p$-Brunn-Minkowski inequality, $p \in [0,1]$, for a general class of measures called convex measures…

We study approximation properties of linear sampling operators in the spaces $L_p$ for $1\le p<\infty$. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of…

Several new inequalities for moduli of smoothness and errors of the best approximation of a function and its derivatives in the spaces $L_p$, $0<p<1$, are obtained. For example, it is shown that for any $0<p<1$ and $k,\,r\in \mathbb{N}$ one…

The $L^p$-Brunn-Minkowski theory for $p\geq 1$, proposed by Firey and developed by Lutwak in the 90's, replaces the Minkowski addition of convex sets by its $L^p$ counterpart, in which the support functions are added in $L^p$-norm.…

In this paper, we establish the existence of smooth, origin-symmetric, strictly convex solutions to the prescribed even $L_p$ curvature problem.

We study the regularity of Lipschitz viscosity solutions to the $\sigma_k$ Yamabe problem in the negative cone case. If either $k=n$ or the manifold is conformally flat and $k>n/2$, we prove that all Lipschitz viscosity solutions are smooth…

We study H\"older continuity of solutions to the Dirichlet problem for measures having density in $L^p$, $p>1$, with respect to Hausdorff-Riesz measures of order $2n-2+\epsilon$ for $0<\epsilon \leq 2$, in a bounded strongly hyperconvex…

We derive quantitative stability results for Minkowski bodies, as well as their counterparts, the $L_p$-Minkowski bodies in the range $1 \le p \neq n$. We prove that, for every pair of probability measures $\mu,\nu$ satisfying a…

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…

The problem of testing hypothesis that a density function has no more than $\mu$ derivatives versus it has more than $\mu$ derivatives is considered. For a solution, the $L^2$ norms of wavelet orthogonal projections on some orthogonal…

A sharp quantitative version of the $L_p-$mixed volume inequality is established. This is achieved by exploiting an improved Jensen inequality. This inequality is a generalization of Pinsker-Csisz\'ar-Kullback inequality for the Tsallis…

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…

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…