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Tangential approximations. (English) Zbl 0613.49016

Let E be a real locally convex (Hausdorff) space. For \(z\in E\) let \({\mathcal N}(z)\) denote a base of neighbourhoods at z. Denote by \({\mathcal K}\) the family of all compact subsets of E. Let \(C\subset E\) and \(x\in C\). The authors define two new tangent cones: \[ F_ C(x):=\cap_{V\in {\mathcal N}(0)}\cup_{X\in {\mathcal N}(x),\lambda >0}\cup_{K\subset V,K\in {\mathcal K}}\cap_{x'\in C\cap X,t\in (0,\lambda)}[t^{-1}(C-x')+K], \]
\[ T^*_ C(x):=\cap_{V\in {\mathcal N}(0)}\cup_{X\in {\mathcal N}(x)}\cap_{x'\in C\cap X,\lambda >0}\cup_{t\in (0,\lambda)}[t^{- 1}(C-x')+V]. \] Some properties of these cones are examined. Next, the class of compactly epi-Lipschitzian sets (which includes all finite- dimensional and all epi-Lipschitzian sets) is introduced. It is proved that if C is compactly epi-Lipschitzian at x, then \(F_ C(x)\) is equal to the Clarke tangent cone \(T_ C(x)\). Some conditions under which \(T^*_ C(x)=T_ C(x)\) are also formulated. Finally, the authors investigate the relationship between \(T_ C(x)\) and the limit inferior of contingent cones at neighbouring points, and prove a theorem which generalizes earlier results of J. P. Penot [ibid. 5, 625-643 (1981; Zbl 0472.58010)] and J. S. Treiman [ibid. 7, 771-783 (1983; Zbl 0515.49013)].
Reviewer: M.Studniarski

MSC:

49J52 Nonsmooth analysis
46A03 General theory of locally convex spaces
46A50 Compactness in topological linear spaces; angelic spaces, etc.
52A99 General convexity
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