Geodesic
Borsuk's problem
Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 621-631.

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The Borsuk number of a bounded set $F$ is the smallest natural number $k$ such that $F$ can be represented as a union of $k$ sets, the diameter of each of which is less than $\operatorname{diam}F$. In this paper we solve the problem of finding the Borsuk number of any bounded set in an arbitrary two-dimensional normed space (the solution is given in terms of the enlargement of a set to a figure of constant width). We indicate spaces for which the solution of Borsuk's problem has the same form as in the Euclidean plane. 
@article{MZM_1977_22_5_a2,
     author = {V. G. Boltyanskii and V. P. Soltan},
     title = {Borsuk's problem},
     journal = {Matemati\v{c}eskie zametki},
     pages = {621--631},
     publisher = {mathdoc},
     volume = {22},
     number = {5},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a2/}
}
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V. G. Boltyanskii; V. P. Soltan. Borsuk's problem. Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 621-631. http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a2/