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.)
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I tried to use Minkowsky's inequality, but it doesn't exactly fit. Jensen's inequality can also not be directly applied. I'm using this to prove a variation of Minkowsky's theorem. – Alt Jan 28 '14 at 02:13
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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.
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1Thanks! besides the norm argument, is there any condition on x and y for which the inequality holds? (for $p>1$) – Alt Jan 28 '14 at 03:28
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