The Dyck and the Preiss separation uniformly

We are concerned with two separation theorems about analytic sets by Dyck and Preiss, the former involves the positively-defined subsets of the Cantor space and the latter the Borel-convex subsets of finite dimensional Banach spaces. We show by introducing the corresponding separation trees that both of these results admit a constructive proof. This enables us to give the uniform version of these separation theorems, and to derive as corollaries the results, which are analogous to the fundamental fact “HYP is effectively bi-analytic” provided by the Suslin–Kleene Theorem.