On Tverberg's conjecture

In 1989 H. Tverberg proposed a quite general conjecture in Discrete geometry, which could be considered as the common basis for many results in Combinatorial geometry and at the same time as a discrete analogue of the common transversal theorems. It implies or contains as the special cases many classical "coincidence" results such as Radon's theorem, Rado's theorem, the Ham-sandwich theorem, the nonembeddability results (of graphs in the plane) etc. The main goal of this note is to verify this conjecture in one new, non-trivial case. We obtain the continuous version of the conjecture. So, it is not surprising that we use topological methods, or more precisely the methods of equivaiant topology and the characteristic classes.