arXiv subjects

Antonín Prochazka

Publications and source records attributed to Antonín Prochazka.

4 recordsLinked to original sources

arXiv2026

On Weak Compactness and Uniform Regularity in Lipschitz free Spaces

We analyze the properties of weakly compact sets in Lipschitz free spaces. Prior research has established that, for a complete metric space $M$, weakly precompact sets in the Lipschitz free space $\mathcal F(M)$ are tight. In this paper, we prove that these sets actually exhibit a stronger property, which we call uniform regularity. However, this condition alone is not sufficient to characterize weakly compact sets, except in the case of scattered metric spaces. On the other hand, if $T$ is an $\mathbb R$-tree, we leverage Godard's isometry between $\mathcal F(T)$ and $L^1(λ_T)$ to obtain an intrinsic characterization of weakly compact sets in $\mathcal F(T)$. This approach allows us to identify conditions that may describe weak compactness across a wider range of spaces. In particular, we provide a characterization of norm-compactness in terms of sums of "large molecules'', while we show that sums of "small molecules'' contain an $\ell_1$-basis.

math.FA↗

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↗

arXiv2024

Delta-points and their implications for the geometry of Banach spaces

We show that the Lipschitz-free space with the Radon--Nikodým property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to $\ell_1$. Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of $\ell_2$, with a $Δ$-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space with a $Δ$-point. Next, we establish powerful relations between existence of $Δ$-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of $Δ$-points for the asymptotic geometry of Banach spaces. In addition, we prove that if $X$ is a Banach space with a shrinking $k$-unconditional basis with $k < 2$, or if $X$ is a Hahn--Banach smooth space with a dual satisfying the Kadets--Klee property, then $X$ and its dual $X^*$ fail to contain $Δ$-points. In particular, we get that no Lipschitz-free space with a Hahn--Banach smooth predual contains $Δ$-points. Finally we present a purely metric characterization of the molecules in Lipschitz-free spaces that are $Δ$-points, and we solve an open problem about representation of finitely supported $Δ$-points in Lipschitz-free spaces.

math.FA↗

arXiv2023

A relative version of Daugavet-points and the Daugavet property

We introduce relative versions of Daugavet-points and the Daugavet property, where the Daugavet-behavior is localized inside of some supporting slice. These points present striking similarities with Daugavet-points, but lie strictly between the notions of Daugavet- and $Δ$-points. We provide a geometric condition that a space with the Radon--Nikodým property must satisfy in order to be able to contain a relative Daugavet-point. We study relative Daugavet-points in absolute sums of Banach spaces, and obtain positive stability results under local polyhedrality of the underlying absolute norm. We also get extreme differences between the relative Daugavet property, the Daugavet property, and the diametral local diameter 2 property. Finally, we study Daugavet- and $Δ$-points in subspaces of $L_1(μ)$-spaces. We show that the two notions coincide in the class of all Lipschitz-free spaces over subsets of $\mathbb{R}$-trees. We prove that the diametral local diameter 2 property and the Daugavet property coincide for arbitrary subspaces of $L_1(μ)$, and that reflexive subspaces of $L_1(μ)$ do not contain $Δ$-points. A subspace of $L_1[0,1]$ with a large subset of $Δ$-points, but with no relative Daugavet-point, is constructed.

math.FA↗