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Mathematics > Metric Geometry

arXiv:2206.01253 (math)
[Submitted on 2 Jun 2022 (v1), last revised 27 Oct 2023 (this version, v2)]

Title:Inscribable order types

Authors:Michael Gene Dobbins, Seunghun Lee
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Abstract:We call an order type inscribable if it is realized by a point configuration where the extreme points are all on a circle. In this paper, we investigate inscribability of order types. We first show that every simple order type with at most 2 interior points is inscribable, and that the number of such order types is $\Theta(\frac{4^n}{n^{3/2}})$. We further construct an infinite family of minimally uninscribable order types. The proof of uninscribability mainly uses Möbius transformations. We also suggest open problems around inscribability.
Subjects: Metric Geometry (math.MG)
Cite as: arXiv:2206.01253 [math.MG]
  (or arXiv:2206.01253v2 [math.MG] for this version)
  https://doi.org/10.48550/arXiv.2206.01253

Submission history

From: Michael Gene Dobbins [view email]
[v1] Thu, 2 Jun 2022 19:08:34 UTC (584 KB)
[v2] Fri, 27 Oct 2023 05:19:47 UTC (561 KB)
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