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 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. The tight span of a metric space can be defined as follows. Let (X,d) be a metric space, and let T(X) 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 For any x, y in X, d(x,y) ≤ f(x) + f(y), and For each x in X, f(x) = sup{d(x,y) - f(y):y in X}. 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. The tight span of (X,d) is the metric space (T(X),δ), where is analogous to the metric induced by the l^∞ norm. (If d is bounded, then δ is the subspace metric induced by the metric induced by the l^∞ norm. If d is not bounded, then every extremal function on X is unbounded and so Regardless, it will be true that for any f,g in T(X), the difference belongs to , i.e., is bounded.) 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(x) = sup{d(x,y) - f(y):y in X}.