Is it true that $|x-y|^n \leq n(|x|^n+ |y|^n)$?

Question

In a lecture note, the author states an inequality without proof. Now, to prove the inequality in his note, one could just prove that

$$|x-y|^n \leq n(|x|^n+ |y|^n)$$

My question is then, is this inequality true? And if so, how to prove it?

The lecture note presented the following inequality The inequality is the following for $+\infty>p\geq 1$: $$ d^p_W(\mu, \nu) = \inf_{\gamma \in \prod(\mu,\nu)} \int_{X \times X} |x-y|^pd\gamma(x,y) \leq p \inf_{\gamma \prod(\mu,\nu)}\int_{X \times X} |x|^p + |y|^p d\gamma (x,y) $$

If the first inequality is not true, than how come is the inequality above true?

I believe the correct form should be $|x-y|^n \le 2^n(|x|^n + |y|^n)$ — player3236, Nov 27 '20 at 21:26
Are there conditions on $p$? E.g. Which $L_p$ are you working in? $1 \leq p \leq 2$? — Calvin Lin, Nov 27 '20 at 21:56

score 2 · Answer 1 · answered Nov 27 '20 at 21:24

2

The inequality is not true.
Take $ x = 1, y = -1, n = 3$ then we have $ 2^3 \leq 3(1+1)$.

It might help for you to state the original inequality.

answered Nov 27 '20 at 21:24

Calvin Lin

68,864

Thanks. I just added the original inequality. Perhaps you can help with it. – Davi Barreira Nov 27 '20 at 21:54

Is it true that $|x-y|^n \leq n(|x|^n+ |y|^n)$?

1 Answers1