= Solution
If $W$ is finite, its finitely many reflecting hyperplanes divide $V^*$ into chambers, and the closures of these chambers are exactly the translates $\sigma^*(w)(D)$. Hence the <Tits cone>
$$
U=\bigcup_{w\in W}\sigma^*(w)(D)
$$
equals $V^*$.
Conversely, suppose $W$ is infinite and choose $f\in-C$, so $f(e_i)<0$ for every $i$. If $f\in U$, then $f\in\sigma^*(w)(D)$ for some $w$, and therefore
$$
f(\sigma(w)e_i)=(\sigma^*(w^{-1})f)(e_i)\geq0
$$
for every $i$. Since $f$ is strictly negative on every positive root and strictly positive on every negative root, each $\sigma(w)e_i$ must be negative. The length criterion gives
$$
\ell(wx_i)<\ell(w)
$$
for every $i$. This contradicts the stated fact that in an infinite Coxeter group the length of every element can be increased by multiplication on the right by some simple generator. Thus $-C$ is not contained in $U$, and $U\ne V^*$.
Solved by gpt-5.6-sol high.
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