How to integrate the maximum norm on $R^n$? 
Let $|x|$ be the maximum norm such that $|x| = \max(|x_1|,...,|x_n|)$ for $x\in \mathbb{R}^{n}$. Calculate the following integral:
$$ \displaystyle\int_{\mathbb{R}^n} \max(0,1-|x|)\ dx.$$

I don't know how to calculate this integral. Maybe I should try to fix
$$x_{ n }\in\left[ -1,1 \right]?$$
Any help?

Here is a hint:
Denote $\|\mathbf{x}\|_\infty:=\max_{1\leq k\leq n}|x_j|$, where $\mathbf{x}=[x_1,\ldots,x_n]^\intercal$.

*

*Consider the sets $E_k=\{\mathbf{x}: \|\mathbf{x}\|_\infty\leq 1, \, |x|_k=\|\mathbf{x}\|_\infty\}$. Notice that this are pairwise disjoint almost surely, that is $m(E_k\cap E_j)=0$ whenever $j\neq k$ where $m$ I Lebesgue's measure.

*By symmetry,  the integral of interest can be rewritten as
$$\int_{\{\|\mathbf{x}\|_\infty\leq1\}}(1-\|\mathbf{x}\|_\infty)\,d\mathbf{x}=\sum^n_{j=1}\int_{E_j}(1-|x_j|)\,d\mathbf{x}=n\int_{E_1}(1-|x_1|)\,d\mathbf{x}$$

*Calculate
$$\int_{E_1}(1-|x|_1)\,d\mathbf{x}=\int^{1}_{-1}(1-|x_1|)2^{n-1}|x_1|^{n-1}\,dx_1$$
