Although this question may have an obvious yes or no answer I was wondering whether in uniformly convex spaces one can take combinations as well in the sense that for normalized $x,y$ with
implies not only that:
but also for any $t \in (\varepsilon',1-\varepsilon')$
$$\Vert tx+(1-t)y \Vert \leq 1-\delta$$
In other words, is there anything special about the midpoint?