  A d-partite hypergraph is called *fractionally balanced* if there exists a
non-negative, not identically zero, function on its edge set that has constant
degrees in each vertex side. Using a topological version of Hall's theorem we
prove lower bounds on the matching number of such hypergraphs. These bounds
yield rainbow versions of the KKM theorem for products of simplices, which in
turn are used to obtain some results on multiple-cake division, and on rainbow
matchings in families of d-intervals.
