The pairing $x \mapsto \sum_s \alpha_s\, x_s$ is continuous in $x$, as a finite sum of continuous functions.
A radical vector $v_0$ (one satisfying $\sum_u v_0(u)\,B(e_u, e_t) = 0$ for all $t$) is fixed by every simple reflection $\sigma_s$ acting on the root space.
A radical vector is fixed by the iterated reflection action $\sigma_{ws}$ for any word $ws$, by induction on the length of $ws$.
Pairing-invariance: the pairing with a radical vector $v_0$ is invariant under the dual word action, since $\sigma$ and its dual are mutually transpose.
Forward direction: a nonzero point of the Tits cone has strictly positive pairing with any strictly-positive radical vector $v_0$.
The set of nonpositive roots at $x$ is invariant under positive scaling of $x$, since the sign of $\langle \alpha, x\rangle$ is preserved.
Reverse direction: if $\langle v_0, x\rangle > 0$ then the nonpositive-roots set at $x$ is finite; we rescale to the hyperplane $\langle v_0, x'\rangle = 1$ and apply finiteness on that hyperplane.
Main characterisation: the Tits cone minus the origin equals the open half-space $\{x : \langle v_0, x\rangle > 0\}$ cut out by any strictly positive radical vector $v_0$ — this is the Perron–Frobenius corollary in the affine Coxeter setting.
A nonzero point of the Tits cone has finitely many positive roots on which it is nonpositive — the nu-finiteness consequence of the previous theorem.
At a nonzero point $x$ of the Tits cone, only finitely many positive roots $\alpha$ satisfy $\langle \alpha, x\rangle = 0$ (a subset of the nu-finite set).
Pointwise local finiteness of the root-hyperplane arrangement: at any nonzero point of the Tits cone, only finitely many root hyperplanes $\{\langle \alpha, y\rangle = 0\}$ pass through it.