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Journal of Convex Analysis 18 (2011), No. 4, 983--997
Copyright Heldermann Verlag 2011

The Role of Local Convexity in Lipschitz Maps

Fernando Albiac
Mathematics Department, Public University of Navarre, 31006 Pamplona-Iruņa, Spain

Lipschitz maps between metric spaces (in particular, between Banach spaces) are abundant and afford a great deal of flexibility: they can be glued, pasted, and truncated without impairing the Lipschitz property. When their target space is the real line, they also can be extended to the whole space without increasing the Lipschitz contstant. However, if we drop the local convexity from the spaces, Lipschitz maps can behave in a completely different way and, in fact, we need not take for granted even their existence; for instance, the Lipschitz-dual of Lp for 0 < p < 1 is trivial, that is to say, there are no nonzero Lipschitz functions f from Lp to the reals R with f(0) = 0. In this note we emphasize the role of local convexity in some properties of Lipschitz maps by showing that local convexity is a necessary condition for these properties to hold, whence they cannot be translated to the nonlocally convex setting.

Keywords: p-convexity, quasi-Banach space, Lipschitz map.

MSC: 46A16, 46T99

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