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 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 finite metric space can be defined as follows. Let (X,d) be a metric space, with X finite, and let T(X) be the set of functions f from X to R such that
In particular (taking x = y in property 1 above) f(x) ≥ 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 (X,d). The second requirement states that none of these distances can be reduced without violating the triangle inequality.
Given two functions f and g in T(X), define δ(f,g) = max |f(x)-g(x)|; if we view T(X) as a subset of a vector space R|X| then this is the usual L∞ distance between vectors. The tight span of X is the metric space (T(X),δ). There is an isometric embedding of X into its tight span, given by mapping any x into the function fx(y) = d(x,y). It is straightforward to expand the definition of δ using the triangle inequality for X to show that the distance between any two points of X is equal to the distance between the corresponding points in the tight span.
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