We prove a number of results involving categories enriched over CMet, the category of complete metric spaces with possibly infinite distances. The category CPMet of path complete metric spaces is locally ${\aleph }_1$-presentable, closed monoidal, and coreflective in CMet. We also prove that the category CCMet of convex complete metric spaces is not closed monoidal and characterize the isometry-${\aleph }_0$-generated objects in CMet, CPMet and CCMet, answering questions by Di Liberti and Rosický. Other results include...

Let T be a precompact subset of a Hilbert space. We estimate the metric entropy of co(T), the convex hull of T, by quantities originating in the theory of majorizing measures. In a similar way, estimates of the Gelfand width are provided. As an application we get upper bounds for the entropy of co(T), $T=t_1,t_2,...$, $\left|\right|t_j\left|\right|\le a_j$, by functions of the $a_j$’s only. This partially answers a question raised by K. Ball and A. Pajor (cf. [1]). Our estimates turn out to be optimal in the case of slowly decreasing sequences ${\left(a_j\right)}_{j=1}^{\infty }$.

For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the metric entropy, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous...

In the first section of this paper there are given criteria for strict convexity and smoothness of the Bochner-Orlicz space with the Orlicz norm as well as the Luxemburg norm. In the second one that geometrical properties are applied to the characterization of metric projections and zero mean valued best approximants to Bochner-Orlicz spaces.

Let X be a closed subspace of c₀. We show that the metric projection onto any proximinal subspace of finite codimension in X is Hausdorff metric continuous, which, in particular, implies that it is both lower and upper Hausdorff semicontinuous.

We describe an approach to establish a theory of metric Sobolev spaces based on Lipschitz functions and their pointwise Lipschitz constants and the Poincaré inequality.

We show that a geodesic metric space which does not admit bilipschitz embeddings into Banach spaces with the Radon-Nikodým property does not necessarily contain a bilipschitz image of a thick family of geodesics. This is done by showing that no thick family of geodesics is Markov convex, and comparing this result with results of Cheeger-Kleiner, Lee-Naor, and Li. The result contrasts with the earlier result of the author that any Banach space without the Radon-Nikodým property contains a bilipschitz...

A metric space (M,d) is said to have the small ball property (sbp) if for every ε₀ > 0 it is possible to write M as the union of a sequence (B(xₙ,rₙ)) of closed balls such that the rₙ are smaller than ε₀ and lim rₙ = 0. We study permanence properties and examples of sbp. The main results of this paper are the following: 1. Bounded convex closed sets in Banach spaces have sbp only if they are compact. 2. Precisely the finite-dimensional Banach spaces have sbp. (More generally: a complete metric...

We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property (umap) in terms of “block unconditionality”. Then we focus on translation invariant subspaces $L_E^p\left(\right)$ and $C_E\left(\right)$ of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces $p_E\left(\right)$, p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between $L_E^p\left(\right)$ and $L_E^{p+2}\left(\right)$. These...

We introduce and study the metric or extreme versions of the notions of a flat and an injective normed module. The relevant definitions, in contrast with the standard known ones, take into account the exact value of the norm of the module. The main result gives a full characterization of extremely flat objects within a certain category of normed modules. As a corollary, some Hahn-Banach type theorems for normed modules are obtained.

Properties of metrically convex functions in normed spaces (of any dimension) are considered. The main result, Theorem 4.2, gives necessary and sufficient conditions for a function to be metrically convex, expressed in terms of the classical convexity theory.

Let ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ be the set of partial isometries with finite rank of an infinite dimensional Hilbert space $\mathcal{H}$. We show that ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ is a smooth submanifold of the Hilbert space ${\mathcal{B}}_2\left(\mathcal{H}\right)$ of Hilbert-Schmidt operators of $\mathcal{H}$ and that each connected component is the set ${\mathcal{I}}_N$, which consists of all partial isometries of rank $N<\mathrm{\infty }$. Furthermore, ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ is a homogeneous space of ${\mathcal{U}}_{\left(\mathrm{\infty }\right)}\times {\mathcal{U}}_{\left(\mathrm{\infty }\right)}$, where ${\mathcal{U}}_{\left(\mathrm{\infty }\right)}$ is the classical Banach-Lie group of unitary operators of $\mathcal{H}$, which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian metrics...

Given a unital C*-algebra $$𝒜$$ and a right C*-module $$𝒳$$ over $$𝒜$$ , we consider the problem of finding short smooth curves in the sphere $${𝒮}_{𝒳}$$ = x ∈ $$𝒳$$ : 〈x, x〉 = 1. Curves in $${𝒮}_{𝒳}$$ are measured considering the Finsler metric which consists of the norm of $$𝒳$$ at each tangent space of $${𝒮}_{𝒳}$$ . The initial value problem is solved, for the case when $$𝒜$$ is a von Neumann algebra and $$𝒳$$ is selfdual: for any element x 0 ∈ $${𝒮}_{𝒳}$$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ $${ℒ}_{𝒜}\left(𝒳\right)$$ , Z* = −Z and ∥Z∥ ≤ π, such...

Proporcionamos una demostración muy corta de un teorema de Cascales-Orihuela que establece que todo conjunto precompacto de un espacio localmente convexo de la clase G (en el sentido de Cascales-Orihuela) es metrizable.

We obtain theorems of metrization and quasi-metrization for several topologies of weak* type on the unit ball of the dual of any separable quasi-normed cone. This is done with the help of an appropriate version of the Alaoglu theorem which is also obtained here.