Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.38856

Characterization of sets in metric spaces where the big Lipschitz derivative is infinite

Jesús Llorente·Oleksandr Maslyuchenko·Ziemowit Wójcicki
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $X$ be a metric space without isolated points. In this paper we prove that $E$ is a $G_δ$-set in $X$ if and only if there exists a continuous function $f:X\to\mathbb{R}$ whose big Lipschitz derivative is infinite precisely on $E$. To this end, we introduce a new generalization of the classical Takagi function to the setting of metric spaces.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jesús Llorente, Oleksandr Maslyuchenko, Ziemowit Wójcicki. 2026-10-28. Characterization of sets in metric spaces where the big Lipschitz derivative is infinite. https://arxiv.org/abs/2610.38856

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On avatars of the Singer-Wermer conjecture in Frechet algebras

We construct two Fréchet algebras admitting countably many mutually inequivalent Fréchet algebra topologies. The second example is a modification of the maiden (and first) example of a non-Banach Fréchet algebra with two inequivalent Fréchet algebra topologies, considered by Read to show that the famous Singer-Wermer conjecture does not hold in the Fréchet case. We discuss a couple of avatars of the Singer-Wermer conjecture in Fréchet algebras. In particular, this conjecture (and so, its weaker version, too) does hold by the second example which also admits countably many mutually equivalent Fréchet algebra topologies. However the stronger version of this conjecture (i.e., the Kaplansky's conjecture) does not hold by the second example.

math.FA↗
arXiv2026

On an affirmative solution to the non-commutative Michael's problem, with applications to automatic continuity theory

In 1952, Michael posed a question about functional continuity of commutative Fréchet algebras in his memoir, known as Michael's problem in the literature. We settle this in the affirmative for the non-commutative Fréchet algebras, along with its equivalent form. Indeed, we continue our previous works to directly attack these problems. We show that the existence of a discontinuous character on the non-commutative test case for this problem -- the Fréchet algebra ${\cal U}_{nc}$ of all entire functions on $\ell^\infty$ -- allows us to generate another Fréchet algebra topology on ${\cal U}_{nc}$, inequivalent to the usual Fréchet algebra topology. An important tool is a topological version of the tensor algebra over a Banach space. The elementary, but crucial, idea is to express the test algebra ${\cal U}_{nc}$ as a {\it weighted Fréchet tensor algebra} over the Banach space $\ell^1$. We give some important applications in automatic continuity theory. In particular, we provide affirmative answers to significant, long-standing problems from 1967 (extending Johnson's result for non-commutative, semi-simple Fréchet algebras; Dales' survey, Question 9), 1976 (extending Esterle's result, giving another approach for Michael's problem for more general $(F)$-algebras), and 1978 (a weaker version of an equivalent form of the non-commutative Michael's problem; Dales' survey, p. 143).

math.FA↗
arXiv2026

Centro-Affine Differential Geometry and the Log-Minkowski Problem

We interpret the log-Brunn-Minkowski conjecture of Böröczky-Lutwak-Yang-Zhang as a spectral problem in centro-affine differential geometry. In particular, we show that the Hilbert-Brunn-Minkowski operator coincides with the centro-affine Laplacian, thus obtaining a new avenue for tackling the conjecture using insights from affine differential geometry. As every strongly convex hypersurface in $\mathbb{R}^n$ is a centro-affine unit-sphere, it has constant centro-affine Ricci curvature equal to $n-2$, in stark contrast to the standard weighted Ricci curvature of the associated metric-measure space, which will in general be negative. In particular, we may use the classical argument of Lichnerowicz and a centro-affine Bochner formula to give a new proof of the Brunn-Minkowski inequality. For origin-symmetric convex bodies enjoying fairly generous curvature pinching bounds (improving with dimension), we are able to show global uniqueness in the $L^p$- and log-Minkowski problems, as well as the corresponding global $L^p$- and log-Minkowski conjectured inequalities. As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body $\bar K$ in $\mathbb{R}^n$, there exists an origin-symmetric convex body $K$ with $\bar K \subset K \subset 8 \bar K$, so that $K$ satisfies the log-Minkowski conjectured inequality, and so that $K$ is uniquely determined by its cone-volume measure $V_K$. If $\bar K$ is not extremely far from a Euclidean ball to begin with, an analogous isometric result, where $8$ is replaced by $1+ε$, is obtained as well.

math.FA↗