Recently, I am studying in combinatorial convexity and related topics. I use the book "Combinatorial Convexity and Algebraic Geometry" (GTM 168) as my main reference. The book is very good, all the proofs of theorems and notes are clear. But I have some questions about the supporting hyperplane of a polarity of a convex body.
In this book, all convex body means a convex compact set $K$ in $\mathbb{R}^n$.
We also assume that the origin $0 \in \mathbb{R}^n$ is contained in the interior of $K$.
The polar body $K^*$ of $K$ is defined to be the intersection $\bigcap_{u \in K} {H_u}^- = \lbrace x \ | \ \langle K,x \rangle \leq 1 \rbrace$, where ${H_u}^-$ is the half-space defined by ${H_u}^- = \lbrace x \ | \ \langle u,x \rangle \leq 1 \rbrace$. By some arguments, we can show that $K^*$ is actully equals to $\bigcap_{u \in \partial K} {H_u}^- $.
In Lemma I.6.6., we want to show that if $x \in \partial K$, then $H_x$ is a supporting hyperplane of $K^*$.
The proof said that if $x \in \partial K$, there is a $\beta_{x} \in \mathbb{R}_{\geq 1}$ such that $H_{\beta_{x}}$ is a supporting hyperplane of $K^*$. I have no question about this fact. And then the author define $\widetilde{K}$ to be the convex hull ${\rm conv} \lbrace \beta_x x \ | \ x \in \partial K \rbrace$ and consider the polarity $\widetilde{K}^*$ of $\widetilde{K}$. By I can't show that $\widetilde{K}^*$ is compact.
I know that the maybe I need to use the lemma: Convex hull of a compact set is compact. But I don't know why the set $\lbrace \beta_x x \ | \ x \in \partial K \rbrace$ is compact. My idea is listed as follow:
I want to show that the map $x \mapsto \beta_x x$ is continuous on $\partial K$. I computed that $\beta_x$ is actually equals to $\frac{1}{1 - \Vert x \Vert \cdot d(K^*, H_x)}$, and then, I have no idea about whether this map is continuous or not. Could you help me? Thank you very much!