Let \(\tau\) (\({\mathcal A})\) denote the set of all convex transversals of a collection \({\mathcal A}\) of convex polytopes. Characterizations for finite \({\mathcal A}\), \({\mathcal B}\) with \(\tau\) (\({\mathcal A})=\tau ({\mathcal B})\) and also for \(\tau\) (\(\tau\) (\({\mathcal A}))=\tau ({\mathcal B})\) are established. In the proofs several distributive lattices involving convex sets are introduced. The structure of these lattices is analyzed.