Distances to the two and three furthest points

Let a compact set $F\subset\mathbb R^n$ contain no less thank points. The function $f_k\colon\mathbb R^n\to\mathbb R$ defined by the formula $f_k(M)=\sup\sum_{i=1} k|MA_i|$, where $A_i\in F$ are distinct points in $F$, is convex. For $k=2$ its minimum is attained at the center of the smallest ball containing $F$ or on a segment passing through this center. For $k=3$ (as well as for any odd $k$) the minimum point of $f_k$ is unique, whereas for even $k$ the domain where $f_k$ attains its minimum can include a segment.