In mathematics, the Kirszbraun theorem in mathematical analysis states that if U is a subset of Euclidean space Eⁿ and

f : U → E^m

is a Lipschitz-continuous map, then there is a Lipschitz-continuous map

F: Eⁿ → E^m

that extends f, and has the same Lipschitz constant as f.

Reference

M. D. Kirszbraun. Uber die zusammenziehenden und Lipschitzchen Transformationen. Fund. Math., (22):77–108, 1934.