ON COMPACT CONNECTED SETS IN BANACH-SPACES

Let E be a separable strictly convex Banach space of dimension at leas t 2. It is shown that there exists a nonempty compact connected set X subset of E such that the nearest point mapping p(X) : E --> 2(E) is n ot single valued on a set of points dense in E. Furthermore, it is pro ved that most (in the sense of the Baire category) nonempty compact co nnected sets X subset of E have the above property. Similar results ho ld for the furthest point mapping.