arXiv subjects

Andrés Quilis

Publications and source records attributed to Andrés Quilis.

12 recordsLinked to original sources

arXiv2026

Universality of Lipschitz quotients and the curve-flat index

We study universality of Lipschitz quotients. First, we modify a construction of Johnson, Lindenstrauss, Preiss and Schechtman to obtain a complete separable metric space that has every complete separable metric space as a Lipschitz quotient. Our main result is in the compact setting, where we prove that no such universal metric space can exist. We deduce this impossibility result by studying the curve-flat index, an ordinal index which provides a measure of the complexity of the curve-fragment structure in a metric space. We show that Lipschitz quotients cannot increase this index in compact domains; while there exist compact spaces with arbitrarily high countable curve-flat index. The main technical part of the paper is dedicated to proving a strong version of the latter fact: for every ordinal $α$ and every compact metric space $M$, there exists a compact metric space $N$ such that the curve-flat quotient of $N$ of order $α$ is almost-isometric to $M$.

math.MG↗

arXiv2026

Affine Approximation in Finite Nagata Dimension and Applications to Lipschitz-free spaces

We show that if $M$ is a metric space of Nagata dimension at most $d$, then there exists an atlas on $M$ modeled on $\mathbb R^d$ such that every Lipschitz map $f:M\to Y$ (with values in an arbitrary Banach space $Y$) can be uniformly approximated by maps that are affine, and thus $\mathcal{C}^1$-smooth, with respect to this atlas. The construction relies on random metric partitions and stochastic retractions inside Lipschitz-free spaces. As an application, we introduce approximate continuous upper gradient $X$-structures (ACUG $X$-structures) on metric spaces and prove that every space of finite Nagata dimension carries an ACUG structure modeled on a superreflexive Banach space. Finally, adapting a proof due to Bourgain, we show that if $M$ has an ACUG superreflexive-structure, then the Lipschitz-free space $\mathcal{F}(M)$ has Pelczyński's property (V*). In particular, at least in the compact case, our result recovers all previously known examples of metric spaces $M$ for which $\mathcal{F}(M)$ has property (V*).

math.FA↗

arXiv2025

On curve-flat Lipschitz functions and their linearizations

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 corresponding metric property of $f$, which is a natural extension of the curve-flatness.

math.FA↗

arXiv2024

Non-uniformly continuous nearest point maps

We construct a Banach space satisfying that the nearest point map (also called proximity mapping or metric projection) onto any 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 every infinite-dimensional separable Banach space is arbitrarily close (in the Banach-Mazur distance) to one satisfying the above conditions.

math.FA↗

arXiv2024

Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and $Δ$-points

We prove that there exists an equivalent norm $\Vert\vert\cdot\vert\Vert$ on $L_\infty[0,1]$ with the following properties: (1) The unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ contains non-empty relatively weakly open subsets of arbitrarily small diameter; (2) The set of Daugavet points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is weakly dense; (3) The set of ccw $Δ$-points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is norming. We also show that there are points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ which are not $Δ$-points, meaning that the space $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ fails the diametral local diameter 2 property. Finally, we observe that the space $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces.

math.FA↗

arXiv2023

(Almost isometric) local retracts in metric spaces

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 $dens(N)=dens(S)$. We also prove that metric spaces which are local retracts (respectively almost isometric local retracts) can be characterised in terms of a condition of extendability of Lipschitz functions (respectively almost isometries) between finite metric spaces. Different examples and counterexamples are exhibited.

math.FA↗

arXiv2023

On projectional skeletons and the Plichko property in Lipschitz-free Banach spaces

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 of all $\mathbb{R}$-trees has the Plichko property witnessed by molecules, and define the concept of retractional trees to generalize this result to a bigger class of metric spaces. Finally, we show that no separable subspace of $\ell_\infty$ containing $c_0$ is an $r$-Lipschitz retract for $r < 2$, which implies in particular that $\mathcal{F}(\ell_\infty)$ is not $r$-Plichko for $r < 2$.

math.FA↗

arXiv2023

Counterexamples in rotundity of norms in Banach spaces

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 these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which moreover fails to be UR (resp. WUR). We prove that these norms can be constructed to be Locally Uniformly Rotund (LUR) in Banach spaces admitting such renormings. Additionally, we obtain that in every Banach space with a LUR norm we can find a LUR renorming which is not URED. These results solve three open problems posed by A.J. Guirao, V. Montesinos and V. Zizler. The norms we construct in this first part are dense. In the last part of this note, we solve a fourth question posed by the same three authors by constructing a $C^\infty$-smooth norm in $c_0$ whose dual norm is not strictly convex.

math.FA↗

arXiv2022

Renormings preserving local geometry at countably many points in spheres of Banach spaces and applications

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. Specifically, on the one hand we construct in every separable Banach space admitting a $C^k$-smooth norm an equivalent norm which is $C^k$-smooth but fails to be uniformly Gâteaux in any direction; and on the other hand we produce in $c_0(Γ)$ for any infinite $Γ$ a $C^\infty$-smooth norm whose ball is dentable but whose sphere lacks any extreme points.

math.FA↗

arXiv2022

On isometric embeddings into the set of strongly norm-attaining Lipschitz functions

In this paper, we provide an infinite metric space $M$ such that the set $\mbox{SNA}(M)$ 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 $\mbox{SNA}(M)$ contains an isometric copy of $c_0$ whenever $M$ is a metric space which is not uniformly discrete. In particular, the latter holds true for infinite compact metric spaces while it does not for proper metric spaces. Some positive results in the non-separable setting are also given.

math.FA↗

arXiv2022

A complete metric space without non-trivial separable Lipschitz retracts

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.

math.FA↗

arXiv2021

Lipschitz retractions and complementation properties of Banach spaces

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 Lipschitz functions. Among our applications we show that the Lipschitz free space $\mathcal{F}(X)$ is a Plichko space whenever $X$ is a Plichko Banach space. Our main results include two examples of metric spaces. The first one $M$ contains two points $\{0,1\}$ such that no separable subset of $M$ containing these points is a Lipschitz retract of $M$. The second example fails the analogous property for arbitrary infinite density. Finally, we introduce the metric version of the concept of locally complemented Banach subspace, and prove some metric analogues to the linear theory.

math.FA↗