Does $|x|^p$ with $0<p<1$ satisfy the triangle inequality on $\mathbb{R}$?

By homogeneity, we try to see whether $|1+t|^p\leq 1+|t|^p$ for all $t$, where $0<p<1$. We first look whether it's the case if $t\geq 0$. Define the maps $f(t):=(1+t)^p-t^p-1$ on $\mathbb R_{\geq 0}$. The derivative is $f'(t)=p((1+t)^{p-1}-t^{p-1})\leq 0$, so $f(t)\leq f(0)=0$ and the inequality holds for $t\geq 0$. To deal with the general case, note that $$|1+t|^p\leq |1+|t||^p\leq 1+||t||^p=1+|t|^p.$$

This inequality is useful to deal with the $L^p$ spaces for $0<p<1$, showing that the map $d(f,g)=\int |f-g|^pd\mu$ is a metric.

Since $\frac{d^2 x^p}{dx^2} = p (p-1) x^{p-2} < 0$, the function $x^p$ is a convex function for $x>0$, therefore for $x,y >0$ the inequality is a special case of Jensen's inequality.

$$ \frac{x^p+y^p}{2} \geq \left(\frac{x+y}{2}\right)^p =\frac{2}{2^p} \frac{(x+y)^p}{2} \geq \frac{(x+y)^p}{2} $$

For negative $x$ or $y$ one has $|x| + |y| \geq |x + y|$. Therefore

$$ \frac{|x|^p+|y|^p}{2} \geq \frac{(|x|+|y|)^p}{2} \geq \frac{|x+y|^p}{2}. $$

(Well, there are already two answers)