$\|\cdot \| = \|\cdot \|_1+\|\cdot \|_2$ is clearly a norm on $\mathbb{R}^2$. I need to show that $\mathbb{R}^2$ with this norm is uniformly convex.
There are various definitions of uniform convexity, but I cannot find a way to use the most appropriate definition and obtain the desired result.
Could anyone please help me?