Related papers: On the infimum of certain functionals

For a Banach space $B$ of functions which satisfies for some $m>0$ $$ \max(\|F+G\|_B,\|F-G\|_B) \ge (\|F\|^s_B + m\|G\|^s_B)^{1/s}, \forall F,G\in B \ (*) $$ a significant improvement for lower estimates of the moduli of smoothness…

Assume that $X$ is a Banach space of measurable functions for which Koml\'os' Theorem holds. We associate to any closed convex bounded subset $C$ of $X$ a coefficient $t(C)$ which attains its minimum value when $C$ is closed for the…

Let $f:X\to X$ be a continuous map on a compact metric space with finite topological entropy. Further, we assume that the entropy map $\mu\mapsto h_\mu(f)$ is upper semi-continuous. It is well-known that this implies the continuity of the…

We provide some new consequences on the Lipschitz numerical radius and index which were introduced recently. More precisely, we give some renorming results on the Lipschitz numerical index, introduce a concept of Lipschitz numerical radius…

We give a necessary and sufficient condition for a difference of convex (DC, for short) functions, defined on a locally convex space, to be Lipschitz continuous. Our criterion relies on the intersections of the "epsilon-subdifferentials of…

Let $\H$ denote the discrete Heisenberg group, equipped with a word metric $d_W$ associated to some finite symmetric generating set. We show that if $(X,\|\cdot\|)$ is a $p$-convex Banach space then for any Lipschitz function $f:\H\to X$…

In this paper we collect several examples of convergence of functions of random processes to generalized functionals of those processes. We remark that the limit is always finitely absolutely continuous with respect to Wiener measure. We…

For each closed, positive (1,1)-current \omega on a complex manifold X and each \omega-upper semicontinuous function \phi on X we associate a disc functional and prove that its envelope is equal to the supremum of all…

We prove that for any $\ell_\infty$-sum $Z = \bigoplus_{i \in [n]} X_i$ of finitely many strictly convex Banach spaces $(X_i)_{i \in [n]}$, an extremeness preserving 1-Lipschitz bijection $f\colon B_Z \to B_Z$ is an isometry, by…

Let $\{f_j\}_{j=1}^n$ and $\{g_k\}_{k=1}^m$ be Parseval p-frames for a finite dimensional Banach space $\mathcal{X}$. Then we show that \begin{align} (1) \quad\quad\quad\quad \log (nm)\geq S_f (x)+S_g (x)\geq -p \log…

We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and…

We prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined…

We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp's theorem and James' compactness theorem, but restricting to sets of functionals determined by geometrical properties.…

We show that a necessary and sufficient condition for a smooth function on the tangent bundle of a manifold to be a Lagrangian density whose action can be minimized is, roughly speaking, that it be the sum of a constant, a nonnegative…

Let $X$ and $Z$ be Banach spaces, $A$ a closed subset of $X$ and a mapping $f:A \to Z$. We give necessary and sufficient conditions to obtain a $C^1$ smooth mapping $F:X \to Z$ such that $F_{\mid_A}=f$, when either (i) $X$ and $Z$ are…

Theorem: Let X and Y be two Banach spaces, Phi: X to Y a continuous, linear, surjective operator, and Psi: X to Y a completely continuous operator with bounded range. Then, one has dim{x in X : Phi(x)=Psi(x)} >= dim(Phi^{-1}(0)). Here dim…

We study a semigroup $\phi$ of linear operators acting on a Banach space $X$ which satisfies the condition $\codim X_0<\infty$, where $X_0=\{x\in X \mid \phi_t(x)\underset{t\to\infty}\longrightarrow 0\}.$ We show that $X_0$ is closed under…

We prove minimax theorems for lower semicontinuous functions defined on a Hilbert space. The main tool is the theory of $\Phi$-convex functions and sufficient and necessary conditions for the minimax equality to hold for $\Phi$-convex…

Let us consider a Riemannian manifold $M$ (either separable or non-separable). We prove that, for every $\epsilon>0$, every Lipschitz function $f:M\rightarrow\mathbb R$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$…

For $\alpha > 0$, the $\alpha$-Lipschitz minorant of a function $f: \mathbb{R} \to \mathbb{R}$ is the greatest function $m : \mathbb{R} \to \mathbb{R}$ such that $m \leq f$ and $|m(s)-m(t)| \le \alpha |s-t|$ for all $s,t \in \mathbb{R}$,…