Strictly convex normed space
= Strictly convex normed space
{title2=$\|x\|=\|y\|=1,\ x\ne y\Rightarrow\|(x+y)/2\|<1$}
A <normed vector space> is strictly convex when distinct unit vectors have their midpoint strictly inside the unit ball. The <Lp space> is strictly convex for $1<p<\infty$: strict convexity of $|z|^p$ gives a strict integrated midpoint inequality for distinct unit functions. The endpoint spaces $L^1$ and $L^\infty$ need not have this property.