Positively curved surfaces with no tangent support plane

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