On circumradii of sets and roughly contractive mappings (Q1409804)

This article deals with the properties and applications of the following geometrical characteristics \[ r_A(S)= \inf_{x\in A}\;\sup_{y\in S}\| x-y\|, \qquad C_A(S)= \Bigl\{x\in A: \sup_{y\in S}\| x-y\|= r_A(S)\Bigr\} \] of bounded sets \(S\) in a normed linear space \(X\); the first is called the relative radius (with respect to \(A\)), the second the relative center set of \(S\) (with respect to \(A\)); if \(A=X\) the word ``relative'' must be omitted or replaced with ``absolute''. The main result is the following inequality \[ \inf_{s\in S}\| z-s\|\leq \frac{n}{n+1} \operatorname {diam} S, \qquad z\in \operatorname {conv} S\setminus S \] that holds in an \(n\)-dimensional normed space \(X\) (this inequality is strict if \(n\geq 2\) and the norm is strictly convex). This result is applied to analyse fixed points of \(r\)-roughly \(k\)-contractive mappings \(T:M\to M\) defined by \(\| Tx-Ty \|\leq k\| x-e \|+r\) for all \(x,y\in M\), where \(k\in (0,1)\) and \(r>0\) are given.