A convex \(d\)-polytope, \(P\), is simple if each vertex is contained in precisely \(d\) facets. Now \(P\) is \(d\)-colorable if its dual graph has that property or, equivalently, if the boundary of its dual is a balanced simplicial sphere. By a result of the reviewer [Math. Z. 240, No. 2, 243–259 (2002; Zbl 1054.05039)] \(P\) is \(d\)-colorable if and only if each 2-face has an even number of vertices. As a first result the authors show the following: For \(P\) a \(d\)-colorable simple \(d\)-polytope let \(X_i\) be a covering of \(P\) by closed sets such that each point is covered at most \(d\) times. Then a connected component of at least one of the \(X_i\) intersects at least two facets of the same color. The proof employs the cohomology of a canonical quasitoric manifold, which was associated with \(P\) by M. W. Davis and T. Januszkiewicz [Duke Math. J. 62, No. 2, 417–451 (1991; Zbl 0733.52006)]. In the \(d\)-colorable case that quasitoric manifold is canonical. There is the following interesting corollary: For any such covering of an arbitrary convex polytope \(Q\) we obtain a connected component of one of the sets in the covering, which intersects at least two distinct \(k\)-faces, for some \(k\). This follows from the first result by taking \(P\) to be the full truncation of \(Q\). Further results are concerned with simple \(d\)-polytopes whose facets admit a \((d+1)\)-coloring.