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Lipschitz Constant

The Lipschitz constant of a map f:X->Y between metric spaces is the least L>=0 for which d_Y(f(x),f(y))<=Ld_X(x,y) for all x,y in X, when such a finite constant exists. Equivalently, it is the supremum of the ratios d_Y(f(x),f(y))/d_X(x,y) over distinct x,y. For a domain with at most one point, the constant is zero.

A map with Lipschitz constant at most one is nonexpansive. On the vector space of real Lipschitz functions vanishing at a fixed base point, the Lipschitz constant is a norm. This function space is used to define the Lipschitz-free space.

See also

Bi-Lipschitz Map, Lipschitz Extension Problem, Lipschitz-Free Space, Lipschitz Function, Metric Space, Nonexpansive Map

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References

Naor, A. Metric Embeddings and Lipschitz Extensions. Lecture notes taken by A. Eskenazis, spring 2015. https://web.math.princeton.edu/~naor/homepage%20files/embeddings_extensions.pdf.

Cite this as:

Weisstein, Eric W. "Lipschitz Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LipschitzConstant.html

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