Distances to the two and three furthest points (Q1360797)

For a natural \(k\), let \(F\) be a compact subset of \(\mathbb{R}^n\) which consists of at least \(k+1\) points. The paper concerns properties of the function \(f_k:R^n\to R\) defined by the formula \[ f_k(x)= \sup\left\{ \sum^k_{i=1} |x-a_i |\mid a_1, \dots, a_k\in F,\;a_i \neq a_j \text{ for } i\neq j \right\}. \] The function \(f_1\) has a unique minimizer, the centre of the smallest ball containing \(F\). The author proves that the centre of this ball is also a minimizer of \(f_2\). Whether this minimizer is or is not unique, it depends on certain property of \(F\) (Theorem 1). Generally, the function \(f_k\) is convex and its set of minimizers is a singleton if \(k\) is odd, and can contain a segment if \(k\) is even (Theorem 2 and Remark).