Theorems of Borsuk-Ulam type for flats and common transversals

In this paper some results on the topology of the space of $k$-flats in ${\mathbb{R}}^n$ are proved, similar to the Borsuk-Ulam theorem on coverings of sphere. Some corollaries on common transversals for families of compact sets in ${\mathbb{R}}^n$, and on measure partitions by hyperplanes, are deduced.