Moduli of uniform convexity for convex sets

Let $C$ be a proper, closed subset with nonempty interior in a normed space $X$. We define four variants of modulus of convexity for $C$ and prove that they all coincide. This result, which is classical and well-known for $C=B_X$ (the unit ball of $X$), requires a less easy proof than the particular case of $B_X$. We also show that if the modulus of convexity of $C$ is not identically null then $C$ is bounded. This extends a result by M.V.~Balashov and D.~Repov\v{s}.