Sets associated with the farthest point problem (Q1337107)

Let \(S\) be a convex compact set in \(E^ 2\). For each cardinal number \(n\), let \(S_ n = \{x \in E^ 2: x\) has exactly \(n\) farthest points in \(S\)\} and \(T_ n = \bigcup_{k \geq n} S_ k\). The authors prove that \(T_ 3\) is countable and that \(T_ 2\) is contractible to a point. Let \(\forall x \in E^ 2\), \(d(x) = \max \{\| s - x\|: s \in S\}\) denote the distance from \(x\) to any of its farthest points of \(S\). Let \(L_ r = \{x \in E^ 2: d(x) = r\}\), \(r \geq r_ s\) denote the level curve where \(r_ s\) is the circumradius of \(S\). Let \(F(x) = \{\sigma \in S: \| \sigma - x\| = d(x)\}\) be the set of farthest points in \(S\) from \(x\). The authors give several properties relating the level curves and the boundary of \(S\). Among other results a) they prove that if \(L_ r \subset S_ 1\) for some \(r > r_ s\), then \(\partial S = \bigcup_{x \in L_ r} F(x)\); b) they give a lower bound \(\rho^*\) such that for all \(r > \rho^* L_ r \subset S_ 1\); c) they prove that if \(S\) is \(C^ 3 \partial S\) and \(L_ r\) have the same evolute. The authors suggest that analogous results may hold not only in \(E^ 2\) but in more general normed linear spaces.