# Proving that $|x|^p,p \geq 1$ is convex

I want to show that $|x|^p,p \geq 1$ is convex, for this i have to prove the inequality $|(1-\lambda )x+\lambda y)|^p \leq (1-\lambda)|x|^p+\lambda |y|^p$ for $\lambda \in (0,1)$ Can anyone prove this inequality? I have proved the convexity using the composition of two convex functions, one of which was increasing but I am interested in a direct proof of this inequality.

• Is this over $\mathbb{R}$ or more generally over $\mathbb{R}^n$, or some inner-product space? – Julien Mar 20 '13 at 12:24
• IT would be great if you prove this for $\mathbb R^n$, but you are welcome to prove this in $\mathbb R$ – Mathematician Mar 20 '13 at 12:26
• I don't suppose you want the proof that uses differentiation. – Gautam Shenoy Mar 20 '13 at 12:27
• no I don't, proof using characterization is not needed, i want a direct proof of this inequality – Mathematician Mar 20 '13 at 12:29
• @julien This is not a direct proof of the inequality – Mathematician Mar 20 '13 at 12:30