Demyanov, V. F.; Pallaschke, D. Point derivations for Lipschitz functions and Clarke’s generalized derivative. (English) Zbl 0916.49013 Appl. Math. 24, No. 4, 465-474 (1997). 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)]. Reviewer: M.Degiovanni (Brescia) Cited in 2 Documents MSC: 49J52 Nonsmooth analysis 52A41 Convex functions and convex programs in convex geometry 26A16 Lipschitz (Hölder) classes Keywords:generalized directional derivative; point derivations; Lipschitz functions Citations:Zbl 0121.10204 PDFBibTeX XMLCite \textit{V. F. Demyanov} and \textit{D. Pallaschke}, Appl. Math. 24, No. 4, 465--474 (1997; Zbl 0916.49013) Full Text: DOI EuDML