$k$-Regular maps into Euclidean spaces and the Borsuk–Boltyanskii problem

The Borsuk–Boltyanskii problem is solved for odd $k$, that is, the minimum dimension of a Euclidean space is determined into which any $n$-dimensional polyhedron (compactum) can be $k$-regularly embedded. A new lower bound is obtained for even $k$.