Curvature of the boundary curve of convex set

Question

I've got a very simple question, but I can't get a rigorous proof. Suppose that $X\subset\mathbb{R}^2$ is a convex set, and suppose further that $\gamma$ is a closed regular curve with image $\partial X$. How to prove that the signed curvature is everywhere non-negative?

Answer 1

Let $P\in X$. At each point of the curve, the curve is on only one side of the tangent, and it is the same side $P$ is on. You can readily translate this into statements about scalar products with normals and so on.