Lemma 13.4.4 (Reciprocity).label In Definition 13.4.3. For every $u\in \mathbb{S}^{n-1}$
\begin{align*}\rho_{K^o}(u)=\frac{1}{h_{K}(u)}\end{align*}
Proof. Let $u\in \mathbb{S}^{n-1},t\geq 0$ then
\begin{align*}tu\in K^{o}\iff \inner{x}{tu}\leq,\fall x\in K\iff th_{K}(u)\leq 1\end{align*}
The largest such $t$ is $1/h_{K}(u)$.$\square$
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