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 \(C=B_X\) $C=B_X$ (the unit ball of X), requires a less easy proof than the particular case of \(B_X.\) $B_X.$ 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. Repovš.