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Introduction x - math.ucla.edu
KIRSZBRAUN-TYPE THEOREMS FOR GRAPHSNISHANT CHANDGOTIA, IGOR PAK, AND MARTIN classicalKirszbraun theoremsays that all 1-Lipschitz functionsf:A Rn,A Rn, with the Euclidean metric have a 1-Lipschitz extension toRn. For metric spacesX,Ywesay thatYisX-Kirszbraunif all 1-Lipschitz functionsf:A Y,A X, have a 1-Lipschitzextension toX. We analyze the case whenXandYare graphs with the usual path metric. Weprove thatZd-Kirszbraun graphs are exactly graphs that satisfies a certainHelly s property. Wealso consider complexity aspects of these results in metric geometry is important for many applications, ranging from discretedifferential geometry to numerical methods. The discrete results are stronger as they typicallyimply the continuous results in the limit. Unfortunately, more often than not, straightforwarddiscretizations fall apart; new tools and ideas are needed to even formulate these extensions; [BS08, Lin02] and [Pak09, 21 24].
KIRSZBRAUN-TYPE THEOREMS FOR GRAPHS NISHANT CHANDGOTIA, IGOR PAK, AND MARTIN TASSY Abstract. The classical Kirszbraun theorem says that all 1-Lipschitz functions f : A!
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