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\begin{lemma}
	Suppose $\mcl{H}_i$ separates $a$ from all the $P_j$ except possibly $P_i$. Then (if $P_i$ are constructed by the regularity lemma) all rainbow simplices contain $a$. This is equivalent to $a$ belonging to the unique bounded region of the arrangement $\{\mcl{H}_i\}$.
\end{lemma}

Suppose $\mcl{H}_i = \{x : xv_i = c_i\}$, $1 \leq i \leq n$.

Could $a$ belong to the region $I \subseteq [n]$?


Then $a \in \bigcap_{i\in I} \mcl{H}_i ^+ \bigcap_{i\notin I} \mcl{H}_i^-$.

Lookat opposite region.

Assume $v_1+\dots+v_n = 0$.



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