Does every convex, nonempty, body have a supporting line at every point on its boundary?
This seems intuitively true to me, but I am having a hard time figuring out how to prove it. I was thinking of using a slicing argument. That is, cut a line through the body and then we have a set of curves which are convex. So more generally, do we need to prove that every convex function has a supporting line at every point?
The supporting hyperplane theorem says a convex set in $\mathbb R^n$ has a supporting hyperplane at every point of its boundary. Is that what you're referring to?