Positively curved surfaces with no tangent support plane

We discuss a characterization of positively curved surfaces with the property that, at each point, the tangent plane to 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 which has convex hull equal to the convex hull of 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.