Let $(W,S)$ be an irreducible Coxeter system with non-degenerate bilinear form $B$ on the Euclidean vector space $V$. The simple root attached to $s\in S$ is denoted by $\alpha_s\in V$. Let $\{\omega_s:s\in S\}$ be the dual basis of $\{\alpha_s:s\in S\}$. In Section 6.8 of Humphreys' book “Reflection groups and Coxeter groups” he defines $C=\{v\in V:\forall s\in S:B(v,\alpha_s)>0\}$ and $D=\overline{C}$ (closure with respect to the Euclidean metric). He claims that $D$ is the convex hull of the vectors $\omega_s$. I can't believe this, since for instance, $2\omega_s$ belongs to $C\subseteq D$. Perhaps he meant conical hull?
What bothers me more is the proof of Proposition 6.8. In the last paragraph, I understand that $N$ has two connected components and both are convex.
Each $\omega_s$ is contained in one of the components. Now he wants to argue with convexity, but why lie all $\omega_s$ in the same component? Here is the excerpt:
Humphreys' (very accurate) errata on his homepage does not cover this part of the book. Sadly, he died last year from Covid.
I tend to avoid all geometric interpretations. Let $M=(\alpha_s,\alpha_t)_{s,t}$ be the Gram matrix of the Coxeter system. We know that $M$ has signature $(n-1,1)$. The dual basis can be expressed as $\omega_s=M^{-1}e_s$ where $e_s$ is the $s$-th standard basis vector. Using that $\det(M)<0$ and every principal minor $\det(M_{ss})>0$, we see that $(\omega_s,\omega_s)=e_sM^{-1}e_s=\det(M_{ss})/\det(M)<0$ (this is basically the lemma before the proposition). We need to show that $vM^{-1}v<0$ for every non-negative integer vector $v\ne 0$. Why is that?