I need to justify my intuition. Let $B_h$ is a $d$-dimensional ball around the origin with radius $h$, $S_h$ be a $d$-dimensional sphere around the origin with radius $h$. It is known from polar coordinates that for a good enough function $f$ $$ \int_{B_h} f(|x|) dx = \int_0^h f(r) |S_h| dr , $$ where $|x|=\sqrt{x_1^2+\dots+x_d^2}$, $|S_h|$ is the surface area of $S_h$.
Let now $|x|_\infty=\max\{|x_1|,\dots,|x_d|\}$, $Q_h = [-h,h]^d$, $P_h$ is the surface of the qube $Q_h$.
Is it true that $$ \int_{Q_h} f(|x|_\infty) dx = \int_0^h f(r) |P_h| dr , $$ where $|P_h|$ is the surface area of $P_h$? Since the Jacobian of this change of variables is quite hard to compute, may there are other ways to establish this.