Extreme points of the unit ball of the space of Lipschitz functions

Farmer, Jeff D.

doi:10.1090/S0002-9939-1994-1195718-7

Extreme points of the unit ball of the space of Lipschitz functions

Author: Jeff D. Farmer
Journal: Proc. Amer. Math. Soc. 121 (1994), 807-813
MSC: Primary 46E15; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9939-1994-1195718-7
MathSciNet review: 1195718
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the space of all Lipschitz functions on a metric space with bounded Lipschitz norm, and give an intrinsic characterization of the extreme points of the unit ball. We briefly discuss some examples of extreme Lipschitz functions, and apply the result to show that if the norm of a Banach space is Gateaux differentiable then extreme functions on any one-dimensional subspace may be canonically extended to extreme functions on the whole space.

[dL] K. de Leeuw, Banach spaces of Lipschitz functions, Studia Math. 21 (1961), 55-66. MR 0140927 (25:4341)
[G] S. Gobzas, Extreme points in Banach spaces of Lipschitz functions, Mathematica 31 (54) (1989), 25-33. MR 1073477 (91j:46032)
[Jo] J. Johnson, Banach spaces of Lipschitz functions and vector-valued Lipschitz functions, Trans. Amer. Math. Soc. 148 (1970), 147-169. MR 0415289 (54:3379)
[Li] J. Lindenstrauss, On nonlinear projections in Banach spaces, Michigan Math. J. 11 (1964), 263-287. MR 0167821 (29:5088)
[Ma] E. Mayer-Wolf, Isometries between Banach spaces of Lipschitz functions, Israel J. Math. 38 (1981), 58-74. MR 599476 (82e:46036)
[Rol1] S. Rolewicz, On extremal points of the unit ball in the Banach space of Lipschitz continuous functions, J. Austral. Math. Soc. Ser. A 41 (1986), 95-98. MR 846774 (87k:46022)
[Rol2] -, On optimal observability of Lipschitz systems, Selected Topics in Operations Research and Mathematical Economics, Lecture Notes in Econom. and Math. Systems, vol. 226, Springer-Verlag, Berlin, 1984, pp. 151-158. MR 758099 (86a:49076)
[Ro] A. K. Roy, Extreme points and linear isometries of the Banach space of Lipschitz functions, Canad. J. Math. 20 (1968), 1150-1164. MR 0236685 (38:4980)
[S] D. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240-272. MR 0161177 (28:4385)