Lemma 13.1.3 (Half Space).label Let $a\in \R^{n}\del \curl{0},c\in\R$ then the half space
\begin{align*}H=\curl{x\in\R^n:\inner{a}{x}\leq c}\end{align*}
is convex.
Proof. Let $x,y\in H,\lam\in[0,1]$ then $\inner{a}{x},\inner{a}{y}\leq c$. By linearity of inner product
\begin{align*}\inner{a}{\lam x+(1-\lam)y}=\lam\inner{a}{x}+(1-\lam)\inner{a}{y}\leq \lam c+(1-\lam)c=c\end{align*}
$\square$
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