# Uniformly convex space and lines

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

$$\|x-y\|\geq\varepsilon$$

implies not only that:

$$\left\|\frac{x+y}{2}\right\|\leq 1-\delta$$

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?

## migrated from mathoverflow.netFeb 25 at 1:05

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• I think this question would fit better on math.stackexchange. (The answer is yes, and this is a good exercise.) – Nik Weaver Feb 21 at 14:50