Is the map $x\mapsto\|x\|_p^p$, defined in $L^p[0,1]$ (for $1<p<\infty$), strictly convex?
Maybe there is a classic inequality that can give the conclusion easily. Let me know if this is the case.
Edit: In the case it is not strictly convex, if we restrict the domain of $x\mapsto\|x\|_p^p$ to be the unit ball in $L^p[0,1]$, the map is strictly convex?