Relative central neighbourhood and antipodal generic points

PDF p. 9: Proposition 3.6, proof

Let $E={\mathrm{\Theta }}_J$ and $u\in \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J{)}^{\circ }$ in the finite-dimensional topology of Tits cone intersections, relative roots and arrangements. There exists an ambient Euclidean ball $B$ centred at $u$ such that $B\cap E\subseteq \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J)$, only finitely many members of ${\mathcal{H}}^J$ meet $B\cap E$, and every ambient root hyperplane meeting $B\cap E$ contains $u$. There are $v,v^{\mathrm{\prime }}\in B\cap E$ on no relative hyperplane with $v^{\mathrm{\prime }}=2u-v$. If $\dim E>0$, choose $v\mathrm{\ne }{v}^{\mathrm{\prime }}$, so $u$ is interior to $[v,v^{\mathrm{\prime }}]$; if $E=0$, take $v=v^{\mathrm{\prime }}=u$. No claim is made that every root hyperplane meeting $B\cap \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}$ contains $u$.

The article, PDF p. 9, Proposition 3.6 proof, asserts a stronger ambient condition on $B\cap \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}$. This item deliberately supplies the correct relative condition on $B\cap {\mathrm{\Theta }}_J$, sufficient for that proposition. The original assertion and an explicit boundary-point obstruction are preserved separately in run-local evidence for independent final assessment. The finite-pair separation input has an independent local inversion-set proof and does not use Proposition 3.6. No original-article assertion is used as proof of a prerequisite.