Tight span

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 Isbell (1964), and it was studied and applied by Holsztyński in the 1960s.

It was later independently rediscovered by Dress (1984) and Chrobak & Larmore (1994); see Chepoi (1997) for this history.

The tight span is one of the central constructions of T-theory.

The tight span of a metric space can be defined as follows.

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 (X,d).

The second requirement states that none of these distances can be reduced without violating the triangle inequality.

The tight span of (X,d) is the metric space (T(X),δ), where

) points into RX, a real vector space of dimension n. On the other hand, if we consider the dimension of T(X) as a polyhedral complex, Develin (2006) 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.

An alternative definition based on the notion of a metric space aimed at its subspace was described by Holsztyński (1968), who proved that the injective envelope of a Banach space, in the category of Banach spaces, coincides (after forgetting the linear structure) with the tight span.

Develin & Sturmfels (2004) 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 Develin & Sturmfels (2004a) that, while the tropical convex hull always contains the tight span, it may not coincide with it.

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.