Geodesic
Theorems of Borsuk-Ulam type for flats and common transversals of families of convex compact sets
Sbornik. Mathematics, Tome 200 (2009) no. 10, pp. 1453-1471

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Several results on the topology of the space of $k$-flats in $\mathbb R^n$ similar to the Borsuk-Ulam theorem on coverings of a sphere and a ball are proved. Some corollaries on common transversals for families of subsets of $\mathbb R^n$ and on measure partitions by hyperplanes are deduced. Bibliography: 22 titles.
Keywords: Borsuk-Ulam theorem, common transversals, Helly-type theorems. 
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     author = {R. N. Karasev},
     title = {Theorems of {Borsuk-Ulam} type for flats and common transversals of families of convex compact sets},
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R. N. Karasev. Theorems of Borsuk-Ulam type for flats and common transversals of families of convex compact sets. Sbornik. Mathematics, Tome 200 (2009) no. 10, pp. 1453-1471. http://geodesic.mathdoc.fr/item/SM_2009_200_10_a2/