On the space of compacta with dense set of points with non-single valued nearest point mapping

Abstract.

The function $p_K: {\Bbb R}^n \to {\rm exp}({\Bbb R}^n)$ called nearest point mapping is well-known: for a given compact set $K \subset {\Bbb R}^n$, p _K associates to each $x \in {\Bbb R}^n$ the set of all points of K closest to x. T. Zamfirescu has shown that the set ${\cal K} _p$ consisting of all compacta K, for which p _K is non-single valued at densely many points, has the complement in ${\rm exp}\, {\Bbb R}^n$ of the first Baire category [8]. We show in this paper that the space ${\cal K}_p$ is homeomorphic to the Hilbert space; moreover, it is contained in ${\rm exp}\, {\Bbb R}^n$ as the pseudo-interior in the Hilbert cube without some point.