Geodesic
Distances to the two and three furthest points
Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 703-708 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{MZM_1996_59_5_a5,
     author = {V. A. Zalgaller},
     title = {Distances to the two and three furthest points},
     journal = {Matemati\v{c}eskie zametki},
     pages = {703--708},
     year = {1996},
     volume = {59},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a5/}
}
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V. A. Zalgaller. Distances to the two and three furthest points. Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 703-708. http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a5/

[1] Kurant R., Robbins G., Chto takoe matematika, Gostekhizdat, M.–L., 1941

[2] Shklyarskii D. O., Chentsov N. N., Yaglom I. M., Geometricheskie neravenstva i zadachi na maksimum i minimum, Nauka, M., 1970