Nonconvexities in Univalence

When a mapping is univalent (one-to-one) on a set is a question which has received considerable study. Much of the recent research has focused on the shape of the set on which the mapping is defined. It has been suggested, in fact, that the set must be convex for univalence to hold. This paper presents conditions under which the set need not be convex. We demonstrate first that the set can be diffeomorphic to a convex set and, second, that the set can have holes inside of it. These somewhat surprising results differ from previous results in that they depend not only upon the Jacobian, but also upon the function value. To provide further insight, we also examine the underlying geometry of the situation.