Abstract
This is an old and very natural question. We shall call f(d) the answer to this problem, and record f (1) = 2, which is trivial. For d = 2 the configuration of four triangles in the margin shows f (2) ≥ 4. There is no similar configuration with five triangles, because from this the dual graph construction, which for our example with four triangles yields a planar drawing of K 4, would give a planar embedding of K 5, which is impossible (see page 67). Thus we have
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References
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Aigner, M., Ziegler, G.M. (2004). Touching simplices. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05412-3_13
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DOI: https://doi.org/10.1007/978-3-662-05412-3_13
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