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Separation by Convex Pseudo-Circles

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Abstract

Let S be a finite set of n points in the plane in general position. We prove that every inclusion-maximal family of subsets of S separable by convex pseudo-circles has the same cardinal \(\left( {\begin{array}{c}n\\ 0\end{array}}\right) +\left( {\begin{array}{c}n\\ 1\end{array}}\right) +\left( {\begin{array}{c}n\\ 2\end{array}}\right) +\left( {\begin{array}{c}n\\ 3\end{array}}\right) \). This number does not depend on the configuration of S and is the same as the number of subsets of S separable by true circles. For a fixed \(k \in \{1,\dots ,n-1\}\), we also count the number of elements in a maximal family of k-subsets of S separable by convex pseudo-circles. This time the number depends on the configuration of S, but it is again equal to the number of k-subsets of S separable by true circles. Thus, it is an invariant of S: it does not depend on the choice of the maximal family. To achieve these results, we introduce a graph that generalizes the dual graph of the order-k Voronoi diagram. The vertices of the graph are the elements of a maximal family of k-subsets of S separable by convex pseudo-circles. In order to count the number of these vertices, we show that the graph is realizable as a triangulation.

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Authors and Affiliations

  1. Université de Haute-Alsace, IRIMAS UR 7499, 18 rue des Frères Lumière, 68200, Mulhouse, France

    Nicolas Chevallier, Augustin Fruchard, Dominique Schmitt & Jean-Claude Spehner

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  1. Nicolas Chevallier
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  2. Augustin Fruchard
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  3. Dominique Schmitt
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  4. Jean-Claude Spehner
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Correspondence to Dominique Schmitt.

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Chevallier, N., Fruchard, A., Schmitt, D. et al. Separation by Convex Pseudo-Circles. Discrete Comput Geom 65, 1199–1231 (2021). https://doi.org/10.1007/s00454-020-00190-3

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  • DOI: https://doi.org/10.1007/s00454-020-00190-3

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