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Show that the closed unit ball in a normed linear space is strictly convex iff the norm is strictly sub additive.

One part is easy strictly sub additive implies strictly convex, but I'm not able to prove the other way.

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According to Wikipedia you have to show that the line segment (except the endpoints) lies inside the unit ball $D$.

Let $x,y \in \partial D$. Let $t \in (0,1)$. Then

$$ \|tx + (1-t)y\| < |t|\|x\| |1-t|\|y\|\le |t| + |1-t| = 1$$

which shows the claim.

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