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The drop theorem, the petal theorem and Ekeland’s variational principle. (English) Zbl 0612.49011
The following three statements are considered: (A) (altered Ekeland’s variational principle). Let \(f: M\to {\mathbb{R}}\cup \{+\infty \}\) be an l.s.c. function on a complete metric space (M,d). Suppose f is bounded below and not everywhere \(+\infty\). Then for any \(\gamma >0\) and any \(x_ 0\in M\) there exists \(a\in M\) such that \(f(a)<f(x)+\gamma d(a,x)\) for all \(x\in M\), \(x\neq a\), and f(a)\(\leq f(x_ 0)-\gamma d(a,x_ 0).\)
(F) (the flower petal theorem). Let X be a complete subset of a metric space (E,d). Let \(x_ 0\in X\) and let \(b\in E\setminus X\), \(r\leq d(b,X)\), \(s=d(b,x_ 0)\). Take any \(\gamma >0\) and denote \(P_{\gamma}(a,b)=\{x\in E:\gamma d(x,a)+d(x,b)\leq d(a,b)\}\). Then there exists \(a\in X\cap P_{\gamma}(x_ 0,b)\) such that \(P_{\gamma}(a,b)\cap X=\{a\}.\)
(D) (the drop theorem). Let C be a complete subset of some normed vector space E, let \(x_ 0\in C\) and let B be a closed ball with centre b and radius \(r<d(b,C)\). Denote \(D(a,B)=\{a+t(b-a):b\in B\), \(t\in [0,1]\}\). Then there exists \(a\in C\cap D(x_ 0,B)\) with \(D(a,B)\cap C=\{a\}.\)
The author gives a short proof of (A) and then proves the implications (A)\(\Rightarrow (F)\Rightarrow (D)\Rightarrow (A)\). Some other geometrical properties of Banach spaces are proved as corollaries from the above-mentioned results.
Reviewer: M.Studniarski

49J52 Nonsmooth analysis
46B20 Geometry and structure of normed linear spaces
47H10 Fixed-point theorems
49J27 Existence theories for problems in abstract spaces
Full Text: DOI
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