An unbounded closed nearly uniformly convex set (Q811675)

Let \(C\) be a nonempty proper closed convex subset of a real Banach space \(X\). \(C\) is daid to be nearly uniformly convex (NUC) (respectively, NUC\('\))) with a center \(a\in C\) if for every \(\varepsilon>0\), there exists a \(\delta\), \(1>\delta>0\), such that for every \(\varepsilon\)- separate sequence in \(C\), \[ \text{co}(x_ n)\cap(a+(1-\delta)(C- a))\neq\emptyset\qquad\hbox{(respectively, } \overline{co}(x_ n)\cap(a+(1-\delta)(C-a))\neq\emptyset\text{).} \] The author showed there is an unbounded nearly uniformly convex set in Hilbert space. This gives an answer of a question of Kutzarova and Rolewicz.