Moduli of uniform convexity for convex sets

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