Shrink the finite relative ball to a central arrangement

Let $B_E(u,r_0)$ be supplied by Finite pair separation yields a finite relative ball, with finite set $\mathcal{K}$ of relative hyperplanes meeting it. For each $K=\ker a\in \mathcal{K}$ not containing $u$, choose ${\delta }_K>0$ such that $B_E(u,{\delta }_K)\cap K=\mathrm{\varnothing }$: in positive dimension take any ${\delta }_K<\mathrm{\mid }a(u)\mathrm{\mid }\mathrm{/}{\sum }_i\mathrm{\mid }a(e_i)\mathrm{\mid }$ in orthonormal coordinates. The triangle inequality bounds $\mathrm{\mid }a(x)-a(u)\mathrm{\mid }$ by $({\sum }_i\mathrm{\mid }a(e_i)\mathrm{\mid })\mathrm{\parallel }x-u\mathrm{\parallel }$, verifying this choice.

Choose $0<r<r_0$ smaller than all these finitely many ${\delta }_K$ (if there are none, only require $r<r_0$). A relative hyperplane meeting the smaller ball belongs to $\mathcal{K}$ and must contain $u$. An ambient root hyperplane either contains $E$, and hence $u$, or has a proper relative trace and so also contains $u$ if it meets this ball. Containment in the cone and finiteness persist under shrinking.