Proposition 13.1.15 (Convexity of Ball).label For any $\norm{\cd}{}$ on $\R^{n}$ and any $r>0$ the closed ball $B_{\norm{\cd}{}}(0,r)=\curl{x\in\R^n:\norm{x}{}\leq r}$ is convex.
Proof. Observe
\begin{align*}\norm{\lam x+(1-\lam)y}{}\leq \lam\norm{x}{}+(1-\lam)\norm{y}{}\leq \lam r+(1-\lam)r=r\end{align*}
$\square$
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