Let $W$ be a subset of $\mathbb{C}^d$ and $co (\overline{W})$ be the closed convex hull of $W$ (here $\overline{W}$ is the closure of $W$ with respect to the topology of $\mathbb{C}^d$).
I don't understand what we mean by the closed convex hull of $W$? Also is it true that
$$co (\overline{W}) \subseteq W ?$$
My goal is to compare
$$\sup\{\|\lambda\|_2,\;\lambda=(\lambda_1,\cdots,\lambda_d) \in W\}$$
and
$$\sup\{\|\lambda\|_2,\;\lambda=(\lambda_1,\cdots,\lambda_d) \in co (\overline{W})\},$$
where
$$\|\lambda\|_2:=\left(\displaystyle\sum_{k=1}^d|\lambda_k|^2\right)^{1/2}.$$
Best Answer
The closed convex hull of a set $W$ is the smallest closed convex set containing $W$. Let $R=\sup\{\|\lambda\|:\lambda\in W\}$. Then the ball around $0$ of radius $R$ is closed and convex and contains $W$. Hence $\mbox{co}(\overline{W})\subset B_R(0)$, so $\sup\{\|\lambda\|:\lambda\in\mbox{co}(\overline{W})\}\leq R$. Since also $W\subset\mbox{co}(\overline{W})$ we find $\sup\{\|\lambda\|:\lambda\in\mbox{co}(\overline{W})\}=R$.