I've been thinking lately about a certain generalization of convexity. I'm sure this concept has been studied, but I don't know what it's called. Could you help me find its name?
Convexity in geometry is usually defined by saying that, for any two points on a surface, the line connecting those points does not pass through the surface. With a well-behaved smooth surface in three dimensions, you could equivalently define it as follows: For any point on the surface, the rest of the surface does not intersect the plane tangent to that point.
I'm curious about the following generalization of that second definition:
Given a smooth surface $S$ in three dimensions, for every point $P$ on $S$, there exists some line which goes through $P$, is tangent to the surface at $P$, and does not intersect $S$.
Locally, this imposes a constraint that prevents "bowls", but allows saddles as well as standard convex shapes. It also seems to impose some global constraints that might be more complicated to describe.
Does this property have a name? Are there further generalizations beyond three dimensions?
Edit: Adding an example.
For example, consider the shape formed at a level set of the sum of two metaballs (shown below). This shape is not convex, but it obeys the above generalized property. For any point on this surface, there is a line tangent to the surface that does not intersect the surface. (the line in question is the line through the point on the surface, which is perpendicular to the plane defined by the centers of the two balls and the point on the surface). The sum of three or more metaballs, or a difference of metaballs, does not necessarily obey this property, as those shapes can have indentations for which any tangent line must intersect the surface.
