Geodesic
Transversals of families of sets in $\mathbb{R}^n$ and a~connection between the~Helly and Borsuk theorems
Sbornik. Mathematics, Tome 79 (1994) no. 1, pp. 93-107

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A criterion is obtained for the existence, given a family of convex sets in $\mathbb{R}^n$, of an $m$-dimensional plane intersecting all members of the family. The results are a generalization of the theorems of Helly, Horn–Klee, and Borsuk. Also presented are applications of these results to the geometry of convex sets and to combinatorics. 
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     author = {V. L. Dol'nikov},
     title = {Transversals of families of sets in $\mathbb{R}^n$ and a~connection between {the~Helly} and {Borsuk} theorems},
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V. L. Dol'nikov. Transversals of families of sets in $\mathbb{R}^n$ and a~connection between the~Helly and Borsuk theorems. Sbornik. Mathematics, Tome 79 (1994) no. 1, pp. 93-107. http://geodesic.mathdoc.fr/item/SM_1994_79_1_a6/