Theorems of Borsuk-Ulam type for flats and common transversals of families of convex compact sets

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.
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