Andrés Quilis

• Polytechnic University of Valencia

We show that several operator ideals coincide when intersected with the class of linearizations of Lipschitz maps. In particular, we show that the linearization $\widehat{f}$ of a Lipschitz map $f:M\to N$ is Dunford–Pettis if and only if it is Radon–Nikodým if and only if it does not fix any copy of $L_{1}$. We also identify and study the correspon...

We show that several operator ideals coincide when intersected with the class of linearizations of Lipschitz maps. In particular, we show that the linearization $\hat{f}$ of a Lipschitz map $f:M\to N$ is Dunford-Pettis if and only if it is Radon-Nikod\'ym if and only if it does not fix any copy of $L_1$. We also identify and study the corresponding...

We construct a Banach space satisfying that the nearest point map (also called proximity mapping or metric projection) onto any non-singleton compact and convex subset is continuous but not uniformly continuous. The space we construct is locally uniformly convex, which ensures the continuity of all these nearest point maps. Moreover, we prove that...

We prove that there exists an equivalent norm $$\left| \left| \left| \cdot \right| \right| \right| $$ · on $$L_\infty [0,1]$$ L ∞ [ 0 , 1 ] with the following properties: The unit ball of $$(L_\infty [0,1],\left| \left| \left| \cdot \right| \right| \right| )$$ ( L ∞ [ 0 , 1 ] , · ) contains non-empty relatively weakly open subsets of arbitrarily sm...

We introduce the notion of (almost isometric) local retracts in metric space as a natural non-linear version of the concepts of locally complemented and almost isometric ideals from Banach spaces. We prove that given two metric spaces $N\subseteq M$ there always exists an almost isometric local retract $S\subseteq M$ with $N\subseteq S$ and $dens(N...

We study projectional skeletons and the Plichko property in Lipschitz-free spaces, relating these concepts to the geometry of the underlying metric space. Specifically, we identify a metric property that characterizes the Plichko property witnessed by Dirac measures in the associated Lipschitz-free space. We also show that the Lipschitz-free space...

We prove that, for every perfect compact Hausdorff space $\Omega$, there exists an equivalent norm $\nnorm{\cdot}$ on $C(\Omega)$ with the following properties: \begin{enumerate} \item The unit ball of $(C(\Omega),\nnorm{\cdot})$ contains non-empty relatively weakly open subsets of arbitrarily small diameter; \item The set of Daugavet points of the...

We prove that, for every perfect compact Hausdorff space $\Omega$, there exists an equivalent norm $|||\cdot|||$ on $C(\Omega)$ with the following properties: 1) The unit ball of $(C(\Omega),|||\cdot|||)$ contains non-empty relatively weakly open subsets of arbitrarily small diameter; 2) The set of Daugavet points of the unit ball of $(C(\Omega),||...

We study projectional skeletons and the Plichko property in Lipschitz-free spaces, relating these concepts to the geometry of the underlying metric space. Specifically, we identify a metric property that characterizes the Plichko property witnessed by Dirac measures in the associated Lipschitz-free space. We also show that the Lipschitz-free space...

We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of...

We develop tools to produce equivalent norms with specific local geometry around infinitely many points in the sphere of a Banach space via an inductive procedure. We combine this process with smoothness results and techniques to solve two open problems posed in the recently published monograph [GMZ22] by A. J. Guirao, V. Montesinos and V. Zizler....

In this paper, we provide an infinite metric space M such that the set SNA(M) of strongly norm-attaining Lipschitz functions does not contain a subspace which is isometric to c_0. This answers a question posed by Antonio Avilés, Gonzalo Martínez Cervantes, Abraham Rueda Zoca, and Pedro Tradacete. On the other hand, we prove that SNA(M) contains an...

We construct a complete metric space $M$ of cardinality continuum such that every non-singleton closed separable subset of $M$ fails to be a Lipschitz retract of $M$. This provides a metric analogue to the various classical and recent examples of Banach spaces failing to have linearly complemented subspaces of prescribed smaller density character.

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more general metric setting fits well with the currently active theory of Lipschitz free spaces and spaces of Lipschi...

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more general metric setting fits well with the currently active theory of Lipschitz free spaces and spaces of Lipschi...