Let $M$ be a compact manifold, and let $M_1,ldots, M_k$ (k>2) be embedded submanifolds. Suppose that $pincap_{i=1}^k M_k$ and that for any subset $S$ of ${1,ldots, k}$ and any $jnotin S$ that $cap_{iin S}M_i$ intersects $M_j$ transversally at $p$.
I believe that in this case the fact that $cap_{i=1}^k M_k$ is nonempty is stable (still true after homotoping each $M_i$ a little bit). Does anyone have a reference for this fact?
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