Dense ambiguous loci and residual cut loci (Q2707503)

A length space \(X\) is a metric space in which every pair of points is joined by some arc of length equal to the distance between the points. Such an arc is called a segment. The cut locus \(C(K)\) of the compact set \(K\) in a length space \(X\) is the set of all points \(x\in X\) such that some segment from \(x\) to \(K\), i.e., a segment whose length minimizes the distance from \(x\) to points in \(K\), is included in no segment from some other point to \(K\). A related set, the ambiguous locus \(A(K)\) associated to the compact set \(K\subset X\) is defined to be the set of all \(x\in X\) at which the nearest point mapping from \(x\) to \(K\) is not single-valued.NEWLINENEWLINENEWLINELet \(X\) be a finite-dimensional real smooth and strictly convex Banach space. It is well-known that the space \(\mathcal C\) of all compact convex sets in \(X\) endowed with the Pompeiu-Hausdorff metric is complete.NEWLINENEWLINENEWLINETheorem 1. For most compact convex sets \(K\subset X\), the ambiguous locus \(A(X\setminus\text{int }K)\) is dense.NEWLINENEWLINENEWLINE(It is proved that the set of all \(K\in {\mathcal C}\), for which \(B(b,r_0)\subset K\) and the nearest point mapping is single-valued at all points of \(B(b,r_0)\), is nowhere dense).NEWLINENEWLINENEWLINEIf \((X,d)\) is a length space and \(K\) a closed set in \(X\), then the multijoined locus \(M(K)\) of \(K\) is defined as the set of all \(x\in X\) admitting at least two segments from \(x\) to \(K\). Obviously, \(A(K)\subset M(K)\).NEWLINENEWLINENEWLINETheorem 2. If \(K\) is a closed set in a locally compact complete length space \((X,d)\) with nonbifurcating geodesics, and \(M(K)\) is dense in \(X\), then \(C(K)\setminus M(k)\) is residual in \(X\).NEWLINENEWLINENEWLINEThen, using these theorems, the author strengthens a result of \textit{S. B. Stechkin} [Rev. Math. Pur. Appl. 8, 5-18 (1963; Zbl 0198.16202)] proving the following:NEWLINENEWLINENEWLINETheorem 3. For most compact \(K\subset\mathbb{R}^d\), the nearest point mapping \(p_K : K\to\mathbb{R}\) is single-valued and injective on a residual subset of \(\mathbb{R}^d\) and \(p_K(\mathbb{R}^d\setminus K)\) is of first category in \(K\).NEWLINENEWLINENEWLINETheorem 4. For most convex bodies \(K\subset\mathbb{R}^d\), the nearest point mapping \(q_K:\text{int }K\to \text{bd}K\), \(q_K=p_{\text{bd}K}\mid \text{int }K\) is single-valued and injective on a residual subset of \(\text{int }K\) and \(q_K(\text{int }K)\) is of first category in \(\text{bd}K\).NEWLINENEWLINEFor the entire collection see [Zbl 0948.00038].