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Linear extension operators between spaces of Lipschitz maps and optimal transport. (English) Zbl 1445.49024
The notion of K-gentle partition of unity is introduced in [J. R. Lee and A. Naor, Invent. Math. 160, 59–95 (2005; Zbl 1074.46004)] and the notion of K-Lipschitz retract is studied in [S. I. Ohta, Positivity 13, 407–425 (2009; Zbl 1198.54048)]. Several authors studied K-gentle partition of unity and K-Lipschitz retract [A. Brudnyi and Y. Brudnyi, Algebra Anal. 19, 106–118 (2007; Zbl 1213.54040); P. Enflo, Acta Math. 130, 309–317 (1973; Zbl 0267.46012); G. Godefroy, North-West. Eur. J. Math. 1, 1–6 (2015; Zbl 1386.46021); G. Godefroy and N. Ozawa, Proc. Amer. Math. Soc. 142, 1681–1687 (2014; Zbl 1291.46013); J. Lindenstrauss ,Michigan Math. J. 11, 263–287 (1964; Zbl 0195.42803); A. Naor and Y. Rabani, Israel J. Math. 219, 115–161 (2017; Zbl 1372.46020); N. J. Kalton, Collect. Math. 55, 171–217 (2004; Zbl 1069.46004)].
The principal objective in this paper is to study a weaker notion related to the Kantorovich-Rubinstein transport distance called K-random projection, and to show that K-random projections can still be used to provide linear extension operators for Lipschitz maps.
MSC:
49Q22 Optimal transportation
47A20 Dilations, extensions, compressions of linear operators
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References:
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