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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?

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migrated from mathoverflow.net Feb 25 at 1:05

This question came from our site for professional mathematicians.

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

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