absolute value to the power of some positive number, is it a norm? Is $|x|^p$ (for constant $p> 0 $) a norm?
In other words does the triangular inequality $|x+y|^p\leq |x|^p+|y|^p $ hold in general?
If not, under what conditions it holds? (e.g $-1 \leq x,y\leq 1$, and/or for $p\geq 1 $, etc.)

Inequality
For $0<p\leq{1}$
$$|x+y|^p\leq|x|^p+|y|^p$$
For $p>1$
$$|x+y|^p\leq~2^{p-1} (|x|^p+|y|^p)$$

Case 1: $0<p\leq{1}$

It is obviously right from the Inequality.

Case 2: $p>1$

We choose $x=0.5,y=0.5,p=2$, then $|x+y|^p=1$, but $|x|^p+|y|^p=0.5$. So $|*|^p$ is not a norm on $\mathbb R$ or $\mathbb C$. You can extend it to all $p>1$.


Attention

(1) You say $-1\leq x,y \leq 1$. This may be some mistakes. Because norm is defined on linear space like $\mathbb R$ or $\mathbb C$
(2) Although $|*|^p$ is not a norm when $p>1$, however it is a quasi-norm because the Inequality.

