In real life, we tend to think of curves as being relative to the physical objects that form them, not as abstract divisions of a two-dimensional space, as in math. In other words, when I see a person's shoulder line, I think of it as the shape of his jacket shoulder, not as the curved line that separates space above and below.
Similarly, we think of hills as convex geological formations, and holes in the ground as concave. Taking the mathematical point of view, where we are simply looking at abstract, divided space, the cave would be convex, not concave, which is perversely ridiculous.