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Strongly unique best approximations and centers in uniformly convex spaces. (English) Zbl 0617.41046
The basic result of the paper is Lemma 2.1: Let X be a uniformly convex space with the modulus of convexity of power type $$q\geq 2$$. Then there is a constant $$d>0$$ such that $\| tx+(1-t)y\|^ q\leq t\| x\|^ q+(1-t)\| y\|^ q-dw_ q(t)\| x-y\|^ q$ for all x,y$$\in X$$ and $$t\in (0,1)$$, where $$w_ q(t)=t(1-t)^ q+(1-t)t^ q.$$ Using this result we prove that if X satisfies the assumptions of Lemma 2.1 and $$M\subset X$$ is a closed convex nonempty subset of X then for every $$x\in X$$ there exists a unique $$m\in M$$ such that $$\| x- m\|^ q\leq \| x-y\|^ q-d\| m-y\|^ q$$ for all $$y\in M$$, where $$d>0$$ is as in Lemma 2.1. In particular we show that it is true when X is the Lebesgue space $$L_ p$$ or the Sobolev space $$H^{m,p}$$ $$(m\geq 0,1<p<\infty)$$. We establish similar results for relative centers and asymptotic centers and we apply them to derive a fixed point theorem for uniformly Lipschitzian mappings.

##### MSC:
 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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##### References:
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