On compact connected sets in Banach spaces

Let $\mathbf {E}$ be a separable strictly convex Banach space of dimension at least 2. It is shown that there exists a nonempty compact connected set $X \subset % \mathbf {E}$ such that the nearest point mapping $p_X:% \mathbf {E}\to 2^{% \mathbf {E}}$ is not single valued on a set of points dense in $\mathbf {E}$. Furthermore, it is proved that most (in the sense of the Baire category) nonempty compact connected sets $X\subset % \mathbf {E}$ have the above property. Similar results hold for the furthest point mapping.