Related papers: On the infimum of certain functionals

Lipschitz constants for the width and diameter functions of a convex body in $\mathbb R^n$ are found in terms of its diameter and thickness (maximum and minimum of both functions). Also, a dual approach to thickness is proposed.

Let $(X,{\mathcal A},\mu)$ be a probability space and let $S\colon X\to X$ be a measurable transformation. Motivated by the paper of K. Nikodem [Czechoslovak Math. J. 41(116) (4) (1991) 565--569], we concentrate on a functional equation…

A condition on a Banach function space $X$ is given under which the Coifman-Fefferman and Fefferman-Stein inequalities on $X$ are equivalent.

We study functorial properties of the spaces $R(X)$, introduced in [Studia Math. 197 (2010), 69-79] as a central tool in the analysis of the Hardy operator minus the identity on decreasing functions. In particular, we provide conditions on…

We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where $\F$ is the class of the integrable, Lipschitz functions on…

We prove that a functional with Lipschitzian derivative, when restricted to suitable spheres centered at a noncritical point, has a unique maximum.

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the…

Let $\mathrm{Lip}_0(X)$ be the space of all Lipschitz scalar-valued functions on a pointed metric space $X$. We characterize the approximation property for $\mathrm{Lip}_0(X)$ with the bounded weak* topology using as tools the tensor…

Recently, in arXiv:2304.07149, a bridge was made between the very active area of spaces of Lipschitz real functions on a metric space and holomorphic functions on an open subset of a Banach space. This was done by introducing and studying…

We show that on separable Banach spaces admitting a separating polynomial, any uniformly continuous, bounded, real-valued function can be uniformly approximated by Lipschitz, analytic maps on bounded sets.

For a Banach lattice $X$, its lattice Sch\"affer constant is defined by: \begin{gather*} \lambda^+(X)=\inf\{\max\{\|x+y\|,\|x-y\|\}\,\colon\,\|x\|=\|y\|=1,x,y\geq{\bf0}\}. \end{gather*} In this paper, we investigate this constant, as well…

In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological…

Let $H$ be a real Hilbert space and $\Phi:H\to H$ be a $C^1$ operator with Lipschitzian derivative and closed range. We prove that $\Phi^{-1}(0)\neq \emptyset$ if and only if, for each $\epsilon>0$, there exist a convex set $X\subset H$ and…

Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ and assume that the Hardy--Littlewood maximal operator satisfies the Fefferman--Stein vector-valued maximal inequality on $X$, and let $q\in[1,\infty)$ and $d\in(0,\infty)$.…

Motivated by the Maximum Theorem for convex functions (in the setting of linear spaces) and for subadditive functions (in the setting of Abelian semigroups), we establish a Maximum Theorem for the class of generalized convex functions,…

We proceed with the study of the Nehari manifold method for functionals in $C^1(X \setminus \{0\})$, where $X$ is a Banach space. We deal now with functionals whose fibering maps have two critical points (a minimiser followed by a…

The Lipschitz space of an infinite (locally-finite) graph is defined as the set of functions on the vertices of the graph such that the differences of the values between adjacent vertices remain bounded. In this paper we prove that this set…

Let ${\mathcal B}(H)$ denote the Banach algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\geq 3$, and let $\mathcal A$ and $\mathcal B$ be subsets of ${\mathcal B}(H)$ which contain all rank one operators.…

In 1994, M. M. Popov [On integrability in F-spaces, Studia Math. no 3, 205-220] showed that the fundamental theorem of calculus fails, in general, for functions mapping from a compact interval of the real line into the lp-spaces for 0<p<1,…

For any $p\in[1,\infty)$, we prove that the set of simple functions taking at most $k$ different values is proximinal in B\"ochner spaces $L^p(X)$ whenever $X$ is a dual Banach space with $w^*$-sequentially compact unit ball. With…