The Lipschitz extension problem asks for conditions on a pair of metric spaces X,Y such that every Y-valued Lipschitz map on a subset of X can be extended to all of X with only a bounded multiplicative loss in the Lipschitz constant. This problem dates back to the work of Kirszbraun and Whitney in the 1930s, and has been intensively investigated in the past two decades. In this talk we will present the main known results on the Lipschitz extension problem, as well as several recent breakthroughs. The techniques used in the proofs are based on geometric, probabilistic and combinatorial arguments.
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