Consider the following inequality, $$3|x_1-x_2|+3|x_2-x_3|+2|x_1-x_3|\leq x_1+2x_2+x_3$$It is sufficient to show that, $$4+|x_1-x_2|+|x_2-x_3|\leq x_1+2x_2+x_3$$$$2+|x_1-x_2|\leq x_1+x_2$$$$1+\max\{x_1,x_2\}\leq 1+\max\{x_1,x_2\}$$Equality only holds when $(x_1,x_2,x_3)$ is $(2,1,2)$ or $(1,2,1)$. Adding the first inequality cyclically and then dividing by $6$ results in the original inequality. Equality only holds when $x_i$ alternates between $1$ and $2$. ($n$ must be even).