Degrees of convex dependence in recursively enumerable vector spaces

Abstract

Let W be a recursively enumerable vector space over a recursive ordered field. We show the Turing equivalence of the following sets: the set of all tuples of vectors in W which are linearly dependent; the set of all tuples of vectors in W whose convex closures contain the zero vector; and the set of all pairs ( X, Y) of tuples in W such that the convex closure of X intersects the convex closure of Y. We also form the analogous sets consisting of tuples with given numbers of elements, and prove similar results on the Turing equivalence of these.