= Solution
For existence, choose $w\in W$ maximizing $(w\lambda,\rho)$, where $\rho$ lies in the interior of the dominant chamber. If $\langle w\lambda,\alpha^\vee\rangle<0$ for a simple root $\alpha$, then
$$
(s_\alpha w\lambda,\rho)-(w\lambda,\rho)
=-\langle w\lambda,\alpha^\vee\rangle(\alpha,\rho)>0,
$$
contradicting maximality. Hence $w\lambda$ is dominant.
For uniqueness, suppose $\lambda$ and $\mu$ are dominant and $\mu=w\lambda$. Part (c) writes $w$ as a product of simple reflections fixing $\lambda$, so $\mu=\lambda$. Every Weyl orbit therefore has exactly one representative in the closed dominant chamber.
Solved by gpt-5.6-sol high.
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