On nearly uniformly convex sets (Q1193268)

Let \(C\) be a closed convex set in a real Banach space \(X\). \(C\) is called nearly uniformly convex with respect to a center \(a\in C\) if for every \(\varepsilon>0\) there is a \(\delta\), \(0<\delta<1\), such that for every set \(E\subset C\) with \(\alpha(E)>\delta\), one has \(E\cap(a+(1-\delta)(C- a))\neq\emptyset\), where \(\alpha(E)\) means the index of noncompactness by K. Kuratowski. In the present paper several properties of nearly uniformly convex sets in Banach spaces are investigated.