 Solution

For $f\in C$, duality givesThe positive-root criterion for Coxeter length saysis a positive root. Its basis coefficients are nonnegative and not all zero, so $f(\sigma (w^{-1})e_i)>0$ and ${\sigma }^{\ast }(w)(C)\subseteq A_i$. If the length decreases, that root is negative and the same calculation gives ${\sigma }^{\ast }(w)(C)\subseteq A_i^{-}$.

If ${\sigma }^{\ast }(w)$ is the identity and $w\ne 1$, choose a left descent $x_i$ from the first letter of a reduced expression for $w$. The preceding result givesalthough $C\subseteq A_i$. The two open half-spaces are disjoint, a contradiction. Thus the Dual geometric representation of a Coxeter group is faithful.

Solved by gpt-5.6-sol high.