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.
Definition 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
δ = ( inf { C ∈ R ≥ 0 : | g ( x ) − f ( x ) | ≤ C for all x ∈ X } ) f , g ∈ T ( X ) = ( ‖ g − f ‖ ∞ ) f , g ∈ T ( X ) {\displaystyle \delta =(\inf\{C\in \mathbb {R} _{\geq 0}:|g(x)-f(x)|\leq C{\text{ for all }}x\in X\})_{f,g\in T(X)}=(\|g-f\|_{\infty })_{f,g\in T(X)}}
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 T ( X ) ⊈ ℓ ∞ ( X ) . {\displaystyle T(X)\not \subseteq \ell ^{\infty }(X).} Regardless, it will be true that for any f,g in T(X), the difference g − f {\displaystyle g-f} belongs to ℓ ∞ ( X ) {\displaystyle \ell ^{\infty }(X)} , i.e., is bounded.)
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(x) = sup{d(x,y) - f(y):y in X}. f is pointwise minimal with respect to the aforementioned first requirement, i.e., for any function g from X to R such that d(x,y) ≤ g(x) + g(y) for all x,y in X, if g≤f pointwise, then f=g.
Basic properties and examples For all x in X, 0 ≤ f ( x ) . {\displaystyle 0\leq f(x).}
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