Definition 13.4.3 (Radial Function).label Let $K$ be a convex body in $\R^{n}$ with $0\in int(K)$. Its radial function is
\begin{align*}\rho_{K}(u)\define\max\curl{t\geq 0:tu\in K}&&u\in \mathbb{S}^{n-1}\end{align*}
then with respect to the Lebesgue measure
\begin{align*}\abs{K}\define vol_{n}(K)=\frac{1}{n}\integral{}{\mathbb{S}^{n-1}}\rho_{K}^{n}(u)d\mu_{\mathbb{S}^{n-1}}\end{align*}
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