Equivalence among various derivatives and subdifferentials of the distance function. (English) Zbl 1035.49019
Let \(X\) be a normed linear space with uniformly Gâteaux differentiable norm and \(C\) a nonempty closed subset of \(X\). The authors study differentiability properties of the distance function \(d_C\) associated with \(C\). For example, it is shown that \(d_C\) is strictly differentiable at \(x\in X\setminus C\) if and only if \(d_C\) is regular at \(x\) in the sense of F. H. Clarke. Further equivalent conditions are established involving various subdifferentials. Moreover, if \(X\) is a Hilbert space, then the Fréchet differentiability of \(d_C\) is characterized, among others, in terms of the metric projection \(P_C\) of \(C\).
The results are based on and complement those obtained earlier by J. M. Borwein, S. P. Fitzpatrick and J. R. Giles [J. Math. Anal. Appl. 128, 512–534 (1987; Zbl 0644.46032)], F. H. Clarke, R. J. Stern and P. R. Wolenski [J. Convex Anal. 2, No. 1–2, 117–144 (1995; Zbl 0881.49008)], S. Fitzpatrick [Bull. Aust. Math. Soc. 22, 291–312 (1980; Zbl 0437.46012)], and others.
The results are based on and complement those obtained earlier by J. M. Borwein, S. P. Fitzpatrick and J. R. Giles [J. Math. Anal. Appl. 128, 512–534 (1987; Zbl 0644.46032)], F. H. Clarke, R. J. Stern and P. R. Wolenski [J. Convex Anal. 2, No. 1–2, 117–144 (1995; Zbl 0881.49008)], S. Fitzpatrick [Bull. Aust. Math. Soc. 22, 291–312 (1980; Zbl 0437.46012)], and others.
Reviewer: W. Schirotzek (Dresden)
Keywords:
uniformly Gâteaux differentiable norm; distance function; strict derivatives; proximal subdifferentials; Fréchet derivativeReferences:
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