Related papers: The $m$th order Orlicz projection bodies

The MICZ-Kepler orbits are the non-colliding orbits of the MICZ Kepler problems (the magnetized versions of the Kepler problem). The oriented MICZ-Kepler orbits can be parametrized by the canonical angular momentum $\mathbf L$ and the Lenz…

We give a characterization of those functions whose all translates are complete in certain Orlicz space $L^{\Phi}(\mathbb{R})$. As a consequence, we identified those discrete sets $\Lambda \subseteq \mathbb{R}$ such that there exists a…

We revisit the problem of estimating the profile (also known as the rarity) in the data stream model. Given a sequence of $m$ elements from a universe of size $n$, its profile is a vector $\phi$ whose $i$-th entry $\phi_i$ represents the…

In the present paper we introduce a certain class of non commutative Orlicz spaces, associated with arbitrary faithful normal locally-finite weights on a semi-finite von Neumann algebra $M.$ We describe the dual spaces for such Orlicz…

We prove the Blaschke-Santal\'o inequality restricted to $n$-gons: the extremal polygons are the affine regular $n$-gons. If either the John or the L\"owner ellipse of a planar $o$-symmetric convex body $K$ is the unit circle about $o$,…

We study the finite element approximation of problems involving the weighted $\Phi$-Laplacian, where $\Phi$ is an $N$-function and the weight belongs to the class $A_\Phi$. In particular, we consider a boundary value problem and an obstacle…

This paper investigates a refinement of Marstrand's projection theorem; more specifically, let $\Pi_t, t\in[0,1]$ be a family of $m$ dimensional subspaces of the Euclidean space $\mathbb{R}^n$ and let $P_t:\mathbb{R}^4\mapsto \Pi_t$ be the…

Ellipsoid fitting is of general interest in machine vision, such as object detection and shape approximation. Most existing approaches rely on the least-squares fitting of quadrics, minimizing the algebraic or geometric distances, with…

First of all, we prove that open mappings in Orlicz-Sobolev classes $W^{1,\phi}_{\rm loc}$ under the Calderon type condition on $\phi$ have the total differential a.e. that is a generalization of the well-known theorems of…

Exploiting spectral properties of symmetric banded Toeplitz matrices, we describe simple sufficient conditions for positivity of a trigonometric polynomial formulated as linear matrix inequalities (LMI) in the coefficients. As an…

We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a…

In Musilak-Orlicz type spaces ${\mathcal S}_{\bf M}$, direct and inverse approximation theorems are obtained in terms of the best approximations of functions and generalized moduli of smoothness. The question of the exact constants in…

Let $G$ be a simple algebraic group with $\mathfrak g=Lie(G)$ and $\mathcal O\subset\mathfrak g$ a nilpotent orbit. If $H$ is a reductive subgroup of $G$ with $Lie(H)=\mathfrak h$, then $\mathfrak g=\mathfrak h\oplus\mathfrak m$, where…

We give necessary and sufficient conditions for the boundedness of the maximal commutators $M_{b}$, the commutators of the maximal operator $[b, M]$ and the commutators of the sharp maximal operator $[b, M^{\sharp}]$ in Orlicz spaces…

This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric…

Studying elliptic measure data problem with strongly nonlinear operator whose growth is described by the means of fully anisotropic $N$-function, we prove the uniqueness for a broad class of measures. In order to provide it, the framework…

Generalizing the slicing inequality for functions on convex bodies from [11], it was proved in [4] that there exists an absolute constant $c$ so that for any $n\in \mathbb N$, any $q\in [0,n-1)$ which is not an odd integer, any…

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a nonnegative self-adjoint operator in $L^2(\mathcal{X})$ satisfying the Davies-Gaffney estimates. Let $\varphi:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such…

What is the optimal way to cut a convex bounded domain $K$ in Euclidean space $(\mathbb{R}^n,|\cdot|)$ into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lov\'asz…

Dual pairs of interior and exterior Hardy spaces associated to a simple closed Lipschitz planar curve are considered, leading to a M\"obius invariant function bounding the norm of the Cauchy transform $\bf{C}$ from below. This function is…