Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1406.3215

Uniformly convex metric spaces

Martin Kell
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit a generalized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simple proof of reflexivity is presented and a weak topology of such spaces is analyzed. This topology called co-convex topology agrees with the usualy weak topology in Banach spaces. An example of a $CAT(0)$-spaces with weak topology which is not Hausdorff is given. This answers questions raised by Monod 2006, Kirk and Panyanak 2008 and Espínola and Fernández-León 2009. In the end existence and uniqueness of generalized barycenters is shown and a Banach-Saks property is proved.

Explore related subjects

math.MGmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Martin Kell. 2014-06-12. Uniformly convex metric spaces. https://doi.org/10.2478/agms-2014-0015

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

KEEP EXPLORING

Related papers

arXiv2026

Space-filling surfaces: sharp Hölder continuous parameterizations from squares to cubes

Following a hint of Semmes, we employ Stong's bijections between integer lattices to construct space-filling surfaces, which are higher-dimensional analogues of space-filling curves. For each $m\geq 2$ we build $α$-Hölder continuous parameterizations $f:[0,1]^m\rightarrow[0,1]^{m+1}$ with sharp exponent $α=m/(m+1)$. In particular, there exist $(2/3)$-Hölder continuous surjections from squares to cubes. This solves Arnold's problem 1988--5.

math.MG↗
arXiv2026

The disjoint disks property for Busemann $G$-spaces

We prove that every finite-dimensional Busemann \(G\)-space of dimension at least five has the disjoint disks property (DDP). For a sufficiently small metric sphere \(L=S(c,r)\), we show that every embedded arc contained in an exact distance level is a homotopical \(Z_2\)-set in \(L\). It follows that \(L\) has the disjoint arc-disk property and the disjoint homotopies property. Daverman's product theorem then gives DDP for \(L\times\mathbb R\), and a local avoidance argument at the center yields DDP for the ambient \(G\)-space. Since finite-dimensional Busemann \(G\)-spaces are generalized manifolds, in dimensions at least five the remaining obstruction to the Busemann conjecture is the resolution problem.

math.MG↗
arXiv2026

Every Compact Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Space

Let $(\mathcal{M},d_{\mathrm{GH}})$ denote the Gromov-Hausdorff space of isometry classes of nonempty compact metric spaces. We prove that every nonempty compact metric space is isometrically embeddable into $(\mathcal{M},d_{\mathrm{GH}})$. More precisely, for every $D>0$ and every nonempty compact metric space $K$ with $\operatorname{diam} K\le D$, we realize the space of all $1$-Lipschitz functions on $K$ with values in $[0,D]$ as a family of metrics on a fixed Cantor space. Under this realization, the Gromov-Hausdorff distance agrees exactly with the uniform distance between functions, and each resulting metric space has diameter at most $76D$. We also construct finite approximations for which the Gromov-Hausdorff distance is given by an exact formula, together with a uniform approximation estimate.

math.MG↗