Positivity 13 (2009), 407–425
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020407-19, published online July 5, 2008
Extending Lipschitz and H¨older maps
between metric spaces
Abstract. We introduce a stochastic generalization of Lipschitz retracts, and
apply it to the extension problems of Lipschitz, H¨older, large-scale Lipschitz
and large-scale H¨older maps into barycentric metric spaces. Our discussion
gives an appropriate interpretation of a work of Lee and Naor.
Mathematics Subject Classiﬁcation (2000). 54C20, 26A16, 53C21.
Keywords. Lipschitz retract, Lipschitz map, H¨older map, large-scale Lipschitz
The extendability of Lipschitz maps is one of the central topics in Banach space
theory. The question is, given two Banach spaces Y and Z, whether there exists
an constant C ≥ 1 such that an arbitrary L-Lipschitz map f : X −→ Z from a
subset X ⊂ Y can be extended to a CL-Lipschitz map
f : Y −→ Z. The estimate
of the constant C is of great interest as well. For example, the asymptotic behavior
of C as the dimensions of Y and Z are increasing, and the relationship between C
and other invariants (e.g., the modulus of convexity or smoothness) are important
As Lipschitz maps make sense between general metric spaces, it is natural and
interesting to ask the same question about nonlinear metric spaces Y and Z.The
most fundamental result is McShane’s classical lemma which asserts that C =1if
Z is the real line, and there are rather recent contributions from the viewpoints of
metric geometry and (the nonlinearization of) the geometry of Banach spaces (see
[9,8,1,12,14] etc.). Also the extension problems for H¨older maps and large-scale
Lipschitz maps receive independent interests (see  and ).
Recently, Lee and Naor  have made a deep progress which has an impact
on both the linear and nonlinear settings. They consider another aspect of the
extension problem by ﬁxing certain X and letting Y be arbitrary, while Z is a
Partly supported by the JSPS fellowship for research abroad.