Let $X$ be a Banach space. Consider the modulus of convexity and the characteristic of convexity $\epsilon_0$. Is the modulus of convexity strictly monotonically increasing in the interval $[\epsilon_0, 2]$.
Yes. $\delta(\epsilon)/\epsilon$ is nondecreasing on $[0,2]$, so once $\delta(\epsilon)$ is nonzero it is strictly increasing. See T. Figiel, On the moduli of convexity and smoothness, Studia Math. 56 (1976), 121–155.