On compact connected sets in Banach spaces

Abstract:

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.