How to prove $||X||^p,$ $p>1$ is strictly convex?

Question

How to prove $||X||^p,$ $p>1$ is strictly convex? Here $X\in\mathbb{R}^d$, and $||x||$ is the $l_2$-norm.

I know how to prove $||X||^p$, $p\geq1$ is convex, for example, using Holder. But how to prove the strict convexity?

Answer 1

One approach is to use the following:

Claim: Let $f:I\to {\mathbb R}$, for $I\subset {\mathbb R}$ an open interval. If $f'$ is strictly increasing on $I$, then $f$ is strictly convex on $I$.

If you need to prove this, you can use the mean value theorem.