The "if" part is trivial. Conversely, assume now that ${\displaystyle X}$ is uniformly convex and that ${\displaystyle x,y}$ are as in the statement, for some fixed ${\displaystyle 0<\varepsilon \leq 2}$. Let ${\displaystyle \delta _{1}\leq 1}$ be the value of ${\displaystyle \delta }$ corresponding to ${\displaystyle {\frac {\varepsilon }{3}}}$ in the definition of uniform convexity. We will show that ${\displaystyle \left\|{\frac {x+y}{2}}\right\|\leq 1-\delta }$, with ${\displaystyle \delta =\min \left\{{\frac {\varepsilon }{6}},{\frac {\delta _{1}}{3}}\right\}}$.

If ${\displaystyle \|x\|\leq 1-2\delta }$ then ${\displaystyle \left\|{\frac {x+y}{2}}\right\|\leq {\frac {1}{2}}(1-2\delta )+{\frac {1}{2}}=1-\delta }$ and the claim is proved. A similar argument applies for the case ${\displaystyle \|y\|\leq 1-2\delta }$, so we can assume that ${\displaystyle 1-2\delta <\|x\|,\|y\|\leq 1}$. In this case, since ${\displaystyle \delta \leq {\frac {1}{3}}}$, both vectors are nonzero, so we can let ${\displaystyle x'={\frac {x}{\|x\|}}}$ and ${\displaystyle y'={\frac {y}{\|y\|}}}$. We have ${\displaystyle \|x'-x\|=1-\|x\|\leq 2\delta }$ and similarly ${\displaystyle \|y'-y\|\leq 2\delta }$, so ${\displaystyle x'}$ and ${\displaystyle y'}$ belong to the unit sphere and have distance ${\displaystyle \|x'-y'\|\geq \|x-y\|-4\delta \geq \varepsilon -{\frac {4\varepsilon }{6}}={\frac {\varepsilon }{3}}}$. Hence, by our choice of ${\displaystyle \delta _{1}}$, we have ${\displaystyle \left\|{\frac {x'+y'}{2}}\right\|\leq 1-\delta _{1}}$. It follows that ${\displaystyle \left\|{\frac {x+y}{2}}\right\|\leq \left\|{\frac {x'+y'}{2}}\right\|+{\frac {\|x'-x\|+\|y'-y\|}{2}}\leq 1-\delta _{1}+2\delta \leq 1-{\frac {\delta _{1}}{3}}\leq 1-\delta }$ and the claim is proved.