Mathematical Reflections 2011, Issue 5 - Problem U205

Problem:
Let $E$ be a vectorial space with norms $\left\|\cdot\right\|_1$ and $\left\|\cdot\right\|_2$. Decide if $\min(\left\|\cdot\right\|_1,\left\|\cdot\right\|_2)$ is a norm.

Proposed by Roberto Bosch Cabrera.

Solution:
The answer is no. Let $E=\mathbb{R}^2, \left\|\underline{x} \right\|_1:=\sqrt{x^2_1+x^2_2}, \left\|\underline{x} \right\|_2:=|x_1|+|x_1-x_2|$. Simple computations show that $\left\|\cdot\right\|_1$ and $\left\|\cdot\right\|_2$ are norms on $\mathbb{R}^2$. Put $\underline{x}=(0,0), \underline{y}=(1,0), \underline{z}=(2,1)$. Then
$$\min(\left\|\underline{x}-\underline{y}\right\|_1,\left\|\underline{x}-\underline{y}\right\|_2) + \min(\left\|\underline{y}-\underline{z}\right\|_1,\left\|\underline{y}-\underline{z}\right\|_2) = \min(1,2) + \min(\sqrt{2},1)=2,$$
but $$\min(\left\|\underline{x}-\underline{z}\right\|_1,\left\|\underline{x}-\underline{z}\right\|_2) = \min(\sqrt{5},3) = \sqrt{5},$$
so $$\min(\left\|\underline{x}-\underline{y}\right\|_1,\left\|\underline{x}-\underline{y}\right\|_2) + \min(\left\|\underline{y}-\underline{z}\right\|_1,\left\|\underline{y}-\underline{z}\right\|_2) < \min(\left\|\underline{x}-\underline{z}\right\|_1,\left\|\underline{x}-\underline{z}\right\|_2).$$