We use the theory of quantization to introduce non-commutative versions of metric on state space and Lipschitz seminorm. We show that a lower semicontinuous matrix Lipschitz seminorm is determined by their matrix metrics on the matrix state spaces. A matrix metric comes from a lower semicontinuous matrix Lip-norm if and only if it is convex, midpoint balanced, and midpoint concave. The operator...

Let X be a linear space. A p-norm on X is a real-valued function on X with 0 < p ≤ 1, satisfying the following conditions: (i) ‖x‖p ≥ 0 and ‖x‖p = 0⇔ x = 0, (ii) ‖αx‖p = |α|p‖x‖p, (iii) ‖x+ y‖p ≤ ‖x‖p +‖y‖p for all x, y ∈ X and all scalars α. The pair (X ,‖,‖p) is called a p-normed space. It is a metric linear space with a translation invariant metric dp defined by dp(x, y)= ‖x− y‖p for all x, ...

If K is a 0-symmetric, bounded, convex body in the Euclidean n-space R (with a fixed origin O) then it defines a norm whose unit ball is K itself (see [12]). Such a space is called Minkowski normed space. The main results in this topic collected in the survey [16] and [17]. In fact, the norm is a continuous function which is considered (in the geometric terminology as in [12]) as a gauge functi...

In 2009 [Schneider 1] obtained stability estimates in terms of the BanachMazur distance for several geometric inequalities for convex bodies in an ndimensional Minkowski space En. A unique feature of his approach is to express fundamental geometric quantities in terms of a single function ρ : B×B→ R defined on the set of all convex bodies B in En. In this paper we show that (the logarithm of) t...

To a finite metric space (X, d) one can associate the so called tight-span T (d) of d, that is, a canonical metric space (T (d), d∞) into which (X, d) isometrically embeds and which may be thought of as the abstract convex hull of (X, d). Amongst other applications, the tight-span of a finite metric space has been used to decompose and classify finite metrics, to solve instances of the server a...

This work addresses the problem of approximating a manifold by a simplicial mesh, and the related problem of building triangulations for the purpose of piecewise-linear approximation of functions. It has long been understood that the vertices of such meshes or triangulations should be “well-distributed,” or satisfy certain “sampling conditions.” This work clarifies and extends some algorithms f...

in the present paper, we introduces the notion of integral type contractive mapping with respect to ordered s-metric space and prove some coupled common fixed point results of integral type contractive mapping in ordered s-metric space. moreover, we give an example to support our main result.

in this paper, we introduce the (g-$psi$) contraction in a metric space by using a graph.let $f,t$ be two multivalued mappings on $x.$ among other things, we obtain a common fixedpoint of the mappings $f,t$ in the metric space $x$ endowed with a graph $g.$

The well-known results about the existence of separable, measurable resp., modifications of stochastic processes (e.g., [4, 5]) are generalized to the case of real valued random fields indexed by a separable, separable and locally convex resp., metric space.

A common fixed point theorem for subcompatible mappings satisfying a generalized contractive condition (in the framework of a convex metric space) is proved and also utilized to derive some invariant approximation results. AMS subject classifications: 41A50, 47H10