1lipschitz

1-Lipschitz, or Lipschitz with constant one, denotes a map that does not expand distances by more than a factor of one. Formally, a function f between metric spaces (X, dX) and (Y, dY) is 1-Lipschitz if for all x, x' in X, dY(f(x), f(x')) ≤ dX(x, x'). This is the case L = 1 in the more general Lipschitz condition dY(f(x), f(y)) ≤ L dX(x, y).

A 1-Lipschitz map is a non-expanding map. Consequently, it is in particular continuous and uniformly continuous. The property is preserved under composition: the composition of two 1-Lipschitz maps is again 1-Lipschitz. Every isometry, which preserves distances exactly, is 1-Lipschitz. Other typical 1-Lipschitz examples include the identity map on any metric space, the metric projection onto a nonempty closed convex subset of a Hilbert space, and the real-valued absolute value function x ↦ |x| on the real line, since ||x| − |y|| ≤ |x − y|. The distance-to-a-set function x ↦ d(x, A) = inf_{a∈A} d(x, a) is also 1-Lipschitz.

In Euclidean spaces, if a differentiable function f: R^n → R^m is 1-Lipschitz, then the operator norm of its Jacobian is almost everywhere at most 1; conversely, a bound on the Jacobian norm implies a Lipschitz constant bound.

Fixed points: unlike strict contractions (L < 1), 1-Lipschitz maps do not necessarily have fixed points, though some do. The Banach fixed-point theorem requires strict contraction for uniqueness and existence on complete spaces.

Applications of the 1-Lipschitz condition include bounding distortions in analysis and geometry, and constraining sensitivity in optimization and machine learning.