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Inscribable Order Types

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Abstract

We call an order type inscribable if it is realized by a point configuration where all extreme points are all on a circle. In this paper, we investigate inscribability of order types. We first construct an infinite family of minimally uninscribable order types. The proof of uninscribability mainly uses Möbius transformations and the Frantz ellipse. We further show that every simple order type with at most two interior points is inscribable, and that the number of such order types is \(\Theta (\frac{4^n}{n^{3/2}})\). We also suggest open problems around inscribability.

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Acknowledgements

We would like to thank Xavier Goaoc for bringing the problem of inscribability to our attention. We are also grateful to Jang Soo Kim for his helpful explanation about plane partitions.

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Correspondence to Michael Gene Dobbins.

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Editor in Charge: Csaba D. Tóth

Dedicated to the memory of Eli Goodman.

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This study was supported from KAIST Advanced Institute for Science-X (KAI-X).

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Dobbins, M.G., Lee, S. Inscribable Order Types. Discrete Comput Geom 72, 704–727 (2024). https://doi.org/10.1007/s00454-023-00591-0

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