Let $K$ be a convex body (i.e a non-empty, convex, compact subset of $\mathbb{R}^n$) such that has the origin as an interior point and $\rho(K,u)=\max\{\lambda \geq 0: \lambda u \in K\}$ be its radial function, I want to prove that $$\operatorname{vol}(K)=\dfrac{1}{n} \int_{\mathbb{S}^{n-1}}\rho(u,K)^n \mathrm{d}u$$
I've thought about some change of variables but I don't see how to do it working with a function that is defined as a maximum. If someone could give a detailed proof I'll be really grateful.