The nearest point mapping is single valued nearly everywhere (Q752527)

The function \(p_ K: {\mathbb{R}}^ d\to 2^{{\mathbb{R}}^ d}\) called metric projection or nearest point mapping is well-known: For a given closed set \(K\subset {\mathbb{R}}^ d\), \(p_ K\) associates to each \(x\in {\mathbb{R}}^ d\) the set of all points of K closest to x. It is known that, \(p_ K\) is single valued almost everywhere and at most points of \({\mathbb{R}}^ d\) [\textit{S. Stechkin}, Rev. Roum. Math. Pures Appl. 8, 5-18 (1963; Zbl 0198.162)], i.e. \(p_ K\) is not single valued on a set of measure zero and first Baire category. We prove that \(p_ K\) is single valued nearly everywhere, i.e. \(p_ K\) is not single valued on a \(\sigma\)-porous set, which implies both preceding assertions. We also establish that, for most compact sets K, \(p_ K\) is not single valued at densely many points. This will not happen, however, if the boundary of K is smooth enough.