Triangle inequality for higher powers

Question

Does an inequality of the following type hold: Let $x_{1},\ldots,x_{n}$ be any reals and $r$ an integer.

$\left|\sum_\limits{i=1}^{n}x_{i}\right|^{r}\leq K_{r}\sum_\limits{i=1}^{n}\left|x_{i}\right|^{r}$

With $K_{r}$ independent of $n$.

Answer 1

Jensen's inequality yields the best possible general constant for any real $r \ge 1$: $$\left|\sum \limits_{i = 1}^n x_i\right|^r = n^r \left|\sum \limits_{i = 1}^n \frac{1}{n}x_i\right|^r \le n^r \sum \limits_{i = 1}^n \frac{1}{n}|x_i|^r = n^{r - 1} \sum \limits_{i = 1}^n |x_i|^r$$