Positively curved surfaces with no tangent support plane

Abstract

We discuss a characterization of positively curved surfaces M M with the property that, at each point, the tangent plane to M M is not a support plane for the entire surface. Such positively curved surfaces with no tangent support plane necessarily have non-empty boundary, and any portion B ⊂ ∂ M B\subset \partial M which has convex hull equal to the convex hull of ∂ M \partial M we call a generating set. This set plays a key role in constructing examples. We give various examples among which there is an embedded topological disk with smallest possible generating set.