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We study convergence of generalized Orlicz energies when the lower growth-rate tends to infinity. We generalize results by Bocea--Mih\u{a}ilescu (Orlicz case) and Eleuteri--Prinari (variable exponent case) and allow weaker assumptions: we…
We revisit entropic formulations of the uncertainty principle for an arbitrary pair of positive operator-valued measures (POVM) $A$ and $B$, acting on finite dimensional Hilbert space. Salicr\'u generalized $(h,\phi)$-entropies, including…
We consider a random field, defined on an integer-valued d-dimensional lattice, with covariance function satisfying a condition more general than summability. Such condition appeared in the well-known Newman's conjecture concerning the…
In this note we present the solution of some isoperimetric problems in open convex cones of $\R^n$ in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin…
In this paper we extend the well-known concentration -- compactness principle of P.L. Lions to Orlicz spaces. As an application we show an existence result to some critical elliptic problem with nonstandard growth.
An abstract convergence theorem for a class of generalized descent methods that explicitly models relative errors is proved. The convergence theorem generalizes and unifies several recent abstract convergence theorems. It is applicable to…
We consider sequences of symmetric $U$-statistics, not necessarily Hoeffding-degenerate, both in a one- and multi-dimensional setting, and prove quantitative central limit theorems (CLTs) based on the use of {\it contraction operators}. Our…
We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body $\Omega \subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial \Omega$. This reduces the study of the Neumann…
In this article, we show that Orlicz-Lorentz spaces $\ell^n_{M,a}$, $n\in\mathbb N$ with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if the norm $\|\cdot\|_{M,a}$ satisfies certain Hardy-type…
Conditions for harmonic analysis in generalized Orlicz spaces have been studied over the past decade. One approach involves the generalized inverse of so-called weak $\Phi$-functions. It featured prominently in the monograph Orlicz Spaces…
We extended the study of the linear fractional self maps (e.g. by Cowen-MacCluer and Bisi-Bracci on the unit balls) to a much more general class of domains, called generalized type-I domains, which includes in particular the classical…
In this paper we present sufficient and necessary conditions for inclusion relation between two weighted Orlicz spaces which complete the Osan\c{c}liol result in 2014. One of the keys to prove our results is to use the norm of the…
The criterion for a point in the unit ball to be a strongly exposed point is given. The necessity and sufficiency conditions for Orlicz-Lorentz spaces to possess strongly exposed property are given. Besides, some useful methods are obtained…
Let $\mathbb{B}_p^N$ be the $N$-dimensional unit ball corresponding to the $\ell_p$-norm. For each $N\in\mathbb N$ we sample a uniform random subspace $E_N$ of fixed dimension $m\in\mathbb{N}$ and consider the volume of $\mathbb{B}_p^N$…
We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some "Poisson-type"…
It is a well-known fact -- which can be shown by elementary calculus -- that the volume of the unit ball in $\mathbb{R}^n$ decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as $n \nearrow…
For some class of mappings, which are generalization of space quasiisometries, an upper estimate for a measure of image of a ball is obtained. As consequence, it is obtained one analog of Schwartz lemma for mappings mentioned above. Results…
We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the $d$-dimensional Euclidean space. When the radius $r$ of the balls is large, this volume can be approximated by a polynomial of…
We re-confirm, for the case of the unit p-ball of R^n, one of recent conjectures of G.Kuperberg on centrally symmetric convex bodies.This conjecture was very recently confirmrd for this particular case by D.A.Gutierrez using polygamma…
We present a short proof of Klartag's central limit theorem for convex bodies, using only the most classical facts about log-concave functions. An appendix is included where we give the proof that thin shell implies CLT. The paper is…