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Point derivations for Lipschitz functions and Clarke’s generalized derivative. (English) Zbl 0916.49013

Let \(X\) be an open subset of a normed space \(E\) and let \(\text{ Lip}(X)\) be the Banach algebra of bounded Lipschitz functions \(f:X \to {\mathbb R}\) endowed with the norm \[ \| f\| = \sup \left\{| f(x)| : x \in X\right\} + \sup \left\{{| f(x)-f(y)| \over \| x-y\| }: x,y \in X, x\neq y\right\} . \] The authors study Clarke’s generalized directional derivative \(\left\{f \mapsto f^{0}(x,v)\right\}\), for fixed \(x \in X\) and \(v \in E\), as a continuous sublinear function on \(\text{ Lip}(X)\). In particular, they establish various relations with the notion of point derivation, as introduced in D. R. Sherbert [Trans. Am. Math. Soc. 111, 240-272 (1964; Zbl 0121.10204)].

MSC:

49J52 Nonsmooth analysis
52A41 Convex functions and convex programs in convex geometry
26A16 Lipschitz (Hölder) classes

Citations:

Zbl 0121.10204
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