In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points...

Fréchet space be locally convex (discussed below). The topology of every Fréchet space is induced by some translation-invariant complete metric. Conversely...

functionals. Fréchet spaces are locally convex topological vector spaces that are completely metrizable (with a choice of complete metric). They are generalizations...

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined...

manifolds and sub-Riemannian manifolds. Any complete and convex metric space is a length metric space (Khamsi & Kirk 2001, Theorem 2.16), a result of Karl...

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function...

convexity Convex cone Convex series Convex metric space Carathéodory's theorem (convex hull) Choquet theory Helly's theorem Holomorphically convex hull Integrally-convex...

this glossary. A caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do not have exactly the same meaning...

general locally convex metric spaces. The Cartan–Hadamard theorem in conventional Riemannian geometry asserts that the universal covering space of a connected...

points Convex polygon, a polygon which encloses a convex set of points Convex polytope, a polytope with a convex set of points Convex metric space, a generalization...

a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is...

Banach space (/ˈbɑː.nʌx/, Polish pronunciation: [ˈba.nax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that...

a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vector space can be "uniquely extended"...

These definitions coincide for subsets of a complete metric space, but not in general. A metric space ( M , d ) {\displaystyle (M,d)} is totally bounded...

space – Locally convex topological vector space that is also a complete metric space Hilbert space – Type of vector space in math K-space (functional analysis)...

vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable...

pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced...

mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the n-dimensional...

strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is...

In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number...

is also a convex metric space with respect to the geodesic distance. A subset of n-dimensional Euclidean space En with its usual flat metric is geodesically...

locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces. Many topological vector spaces are spaces of functions...

(Tychonoff) fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : X →...

a metric projection is a function that maps each element of a metric space to the set of points nearest to that element in some fixed sub-space. Formally...

A space with this metric is called Minkowski space. The hypersurface F 0 {\displaystyle F_{0}} is convex and can be irregular. The defined metric is...

also known as a Helly family. A convex metric space in which the closed balls have the 2-Helly property (that is, a space with Helly dimension 1, in the...

of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental,...

points in a metric space is the largest distance between points in the set. As an important special case, the diameter of a metric space is the largest...

Another misnomer is Minkowski metric, but Minkowski space is not a metric space. The group of transformations for Minkowski space that preserves the spacetime...

\varphi (t)=\min(t,1).} Such a metric is called Lévy-metric for L 0 . {\displaystyle L^{0}.} Under this metric the space L 0 {\displaystyle L^{0}} is complete...