Related papers: A non-existence result for the $L_p$-Minkowski pro…

Let $\mu_p$ be the generalized Gaussian distribution on $\mathbb{R}^n$ with density $e^{-\frac{|x|^p}{p}}$ multiplied by a constant depending on $p\ge 1$ and $n$, and $\alpha_p(n)$ be the largest number such that the Brunn-Minkowski type…

In a paper published in 2020 in Studia Mathematica, Abrahamsen et al. proved that in the real space $L_1(\mu)$, where $\mu$ is a non-zero $\sigma$-finite (countably additive non-negative) measure, norm-one elements in finite convex…

This paper is concerned with the $L^p$ integrability of $N$-harmonic functions with respect to the standard weights $(1-|x|^2)^{\alpha}$ on the unit ball $\mathbb{B}$ of $\mathbb{R}^n$, $n\geq 2$. More precisely, our goal is to determine…

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 a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space $\mathcal{D}^{1,p}_0$ into $L^q$ and the summability properties of the distance function. We prove that in the…

We solve the classical problem of Plateau in every metric space which is $1$-complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual…

The Minkowski content $L_0(G)$ of a body $G\subset{\mathbb{R}}^d$ represents the boundary length (for $d=2$) or the surface area (for $d=3$) of $G$. A method for estimating $L_0(G)$ is proposed. It relies on a nonparametric estimator based…

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in…

This paper studies complete non-compact smooth metric measure space $(M^n,g,\mathrm{e}^{-f}\mathrm{d}v)$ with positive first spectrum $\lambda_1(\Delta_f)$ or satisfying a weighted Poincar\'e inequality with weight function $\rho$. We…

We prove that a homeomorphism of a compact metric space has an expansive measure \cite{ms} if and only if it has many ones with invariant support. We also study homeomorphisms for which the expansive measures are dense in the space of Borel…

This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions,…

By using Bernstein-type inequality we define analogs of spaces of entire functions of exponential type in $L_{p}(X), 1\leq p\leq \infty$, where $X$ is a symmetric space of non-compact. We give estimates of $L_{p}$-norms, $1\leq p\leq…

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

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$. We prove that for every symmetric convex sets $K,L$ in $\mathbb{R}^n$ and every $\lambda\in(0,1)$, $$\gamma_n(\lambda K+(1-\lambda)L)^{\frac{1}{n}} \geq \lambda…

Let $\mathbb{R}^n$ be the n-dimensional Euclidean space with $O$ as the origin. Let $\wedge$ be a lattice of determinant $1$ such that there is a sphere $|X|<R$ which contains no point of $\wedge$ other than $O$ and has $n$ linearly…

We study the holomorphic embedding problem from a compact strongly pseudoconvex real algebraic hypersurface into a sphere of higher dimension. We construct a family of compact strongly pseudoconvex hypersurfaces $M_{\epsilon}$ in…

The celebrated Dvoretzky theorem asserts that every $N$-dimensional convex body admits central sections of dimension $d = \Omega(\log N)$, which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to…

We define a measure of noncompactness $\lambda$ on the standard Hilbert $C^*$-module $l^2(\mathcal A)$ over a unital $C^*$-algebra, such that $\lambda(E)=0$ if and only if $E$ is $\mathcal A$-precompact (i.e.\ it is $\varepsilon$-close to a…

For each $p>n$ we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the Sobolev space $W_p^1(R^n)$ to an arbitrary closed subset of $R^n$.

Let $G$ be a noncompact connected Lie group, denote with $\rho$ a right Haar measure and choose a family of linearly independent left-invariant vector fields $\mathbf{X}$ on $G$ satisfying H\"ormander's condition. Let $\chi$ be a positive…