Let D be a closed region in $R^3$ whose boundary S is a closed, smooth, orientable surface. The tangent plane at every point on S intersects it in a connected set $M$. Is D (the interior of S) a convex set?
This statement seems intuitively plausible to me. If it is not true, a counterexample would be welcome. If false, "technical" modifications to the statement that make it true would also be most helpful.
Also, if true, to what extent can the smoothness restriction be weakened / generalized?
Thanks.