I've been reading K. Goebel's paper "An elementary proof of the fixed-point theorem of Browder and Kirk" and the following part has been baffling me.
The Banach space is uniformly convex if and only if there exists an increasing, positive function $\delta:(0,2] \to (0,1]$, such that the inequalities $\|x\| \leq r$, $\|y\|\leq r$, $\|x-y\|\geq r\varepsilon$ imply that \begin{equation}\label{uniform} \left\|\frac{x+y}{2}\right\|\leq r(1-\delta(\varepsilon)).\quad\quad\quad(1) \end{equation}
I know, that a Banach space is uniformly convex when the modulus of convexity is positive. How to show that there exist an INCREASING function $\delta$ satisfying (1)?
EDIT Found the solution in "Geometric Properties of Banach Spaces and Nonlinear Iterations" by Charles Chidume.