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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

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