Consider subsets of $\mathbb{R}^n$. Instead of a straight line, every pair of points have to be joined by a line with the curvature of a parabola or less than it . This would include the straight line too. I guess one could arrive at some differentiability condition for the boundary curve in such cases. Is there something of this kind?
Edit: I just want the absolute value of curvature of the line joining it to be bound in some sense. I am not very particular about it being a parabola.