Tight span


In metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes known as the injective envelope or hyperconvex hull of M. It has also been called the injective hull, but should not be confused with the injective hull of a module in algebra, a concept with a similar description relative to the category of R-modules rather than metric spaces.
The tight span was first described by, and it was studied and applied by Holsztyński in the 1960s. It was later independently rediscovered by and ; see for this history. The tight span is one of the central constructions of T-theory.

Definition

The tight span of a metric space can be defined as follows. Let be a metric space, and let T be the set of extremal functions on X, where we say an extremal function on X to mean a function f from X to R such that
  1. For any x, y in X, d ≤ f + f, and
  2. For each x in X, f = sup.
In particular f ≥ 0 for all x. One way to interpret the first requirement above is that f defines a set of possible distances from some new point to the points in X that must satisfy the triangle inequality together with the distances in. The second requirement states that none of these distances can be reduced without violating the triangle inequality.
The tight span of is the metric space , where
is analogous to the metric induced by the norm.

Equivalent definitions of extremal functions

For a function f from X to R satisfying the first requirement, the following versions of the second requirement are equivalent:
  • For each x in X, f = sup.f is pointwise minimal with respect to the aforementioned first requirement, i.e., for any function g from X to R such that d ≤ g + g for all x,y in X, if g≤f pointwise, then f=g.

Basic properties and examples

  • For all x in X,
  • For each x in X, is extremal.
  • If X is finite, then for any function f from X to R that satisfies the first requirement, the second requirement is equivalent to the condition that for each x in X, there exists y in X such that f + f = d.
  • Say |X|=2, and choose distinct a, b such that X=. Then is the convex hull of .
  • Every extremal function f on X is Katetov: f satisfies the first requirement and or equivalently, f satisfies the first requirement and, or equivalently, f satisfies the first requirement and T⊆C(X). T is equicontinuous.
  • Not every Katetov function on X is extremal. For example, let a, b be distinct, let X =, let d = x,y in X be the discrete metric on X, and let f =. Then f is Katetov but not extremal.
  • If d is bounded, then every f in T is bounded. In fact, for every f in T,
  • If d is unbounded, then every f in T is unbounded.
  • is closed under pointwise limits. For any pointwise convergent
  • If ' is compact, then ' is compact. is a bounded subset of C. We have shown T is equicontinuous, so the Arzelà–Ascoli theorem implies that T is relatively compact. However, the previous bullet implies T is closed under the norm, since convergence implies pointwise convergence. Thus T
  • For any function g'' from X to R that satisfies the first requirement, there exists f in T such that f≤g pointwise.
  • For any extremal function f on X,
  • For any f,g in T, the difference belongs to, i.e., is bounded.
  • The Kuratowski map is an isometry.
  • Let f in T. For any a in X, if f=0, then f=e.
  • ' is hyperbolic if and only if ' is hyperbolic.

Hyperconvexity properties

' and are both hyperconvex.

Examples

  • Say |X|=3, choose distinct a, b, c such that X=, and let i=d, j=d, k=d. Then where
Image:Orthogonal-convex-hull.svg|thumb|If a set of points in the plane, with the Manhattan metric, has a connected orthogonal convex hull, then that hull coincides with the tight span of the points.
  • The figure shows a set X of 16 points in the plane; to form a finite metric space from these points, we use the Manhattan distance. The blue region shown in the figure is the orthogonal convex hull, the set of points z such that each of the four closed quadrants with z as apex contains a point of X. Any such point z corresponds to a point of the tight span: the function f corresponding to a point z is f = d. A function of this form satisfies property 1 of the tight span for any z in the Manhattan-metric plane, by the triangle inequality for the Manhattan metric. To show property 2 of the tight span, consider some point x in X; we must find y in X such that f+''f=d''. But if x is in one of the four quadrants having z as apex, y can be taken as any point in the opposite quadrant, so property 2 is satisfied as well. Conversely it can be shown that every point of the tight span corresponds in this way to a point in the orthogonal convex hull of these points. However, for point sets with the Manhattan metric in higher dimensions, and for planar point sets with disconnected orthogonal hulls, the tight span differs from the orthogonal convex hull.

Dimension of the tight span when ''X'' is finite

The definition above embeds the tight span T of a set of n points into RX, a real vector space of dimension n. On the other hand, if we consider the dimension of T as a polyhedral complex, showed that, with a suitable general position assumption on the metric, this definition leads to a space with dimension between n/3 and n/2.

Alternative definitions

An alternative definition based on the notion of a metric space aimed at its subspace was described by, who proved that the injective envelope of a Banach space, in the category of Banach spaces, coincides with the tight span. This theorem allows to reduce certain problems from arbitrary Banach spaces to Banach spaces of the form C, where X is a compact space.
attempted to provide an alternative definition of the tight span of a finite metric space as the tropical convex hull of the vectors of distances from each point to each other point in the space. However, later the same year they acknowledged in an Erratum that, while the tropical convex hull always contains the tight span, it may not coincide with it.

Applications