lengthlike

Lengthlike is a term used in metric geometry to describe a function that generalizes the notion of the length of a curve between two points. Let X be a set and L: X × X → [0, ∞) a function. L is called lengthlike if it satisfies: L(x, x) = 0 for all x; symmetry L(x, y) = L(y, x); L(x, z) ≤ L(x, y) + L(y, z) for all x, y, z; and L(x, y) > 0 whenever x ≠ y. Furthermore, L is compatible with polygonal paths, in that for any finite sequence x0, x1, ..., xn, L(x0, xn) ≤ ∑ i L(xi, xi+1).

These axioms mirror the basic properties of curve length in Euclidean spaces. When d is a length metric (an intrinsic metric) on X, d itself is lengthlike, and L can be taken to be d. In general, a lengthlike function behaves like a distance that can be approximated by finite chains of jumps.

Examples: The usual Euclidean distance in R^n is lengthlike. A discrete metric with L(x, y) = 1 for x ≠ y is also lengthlike. A function that violates positivity or the triangle inequality is not lengthlike.

Relation to related notions: Lengthlike is connected to length metrics or path metrics, Lipschitz structure, and the study of metric spaces in geometric analysis and geometric group theory. It provides a convenient language for discussing how distances arise from or bound the lengths of connecting paths.

Terminology varies: lengthlike is not a universally standardized term; some authors use it informally to describe any function with length-compatibility properties. The concept helps compare different distance notions and to study intrinsic geometry.