Abstract
A collection \(S=\{S_i, \ldots , S_n\}\) of disjoint closed convex sets in \(\mathbb {R}^d\) is separable if there exists a direction (a non-zero vector) \( \overrightarrow{v}\) of \(\mathbb {R}^d\) such that the elements of S can be removed, one at a time, by translating them an arbitrarily large distance in the direction \( \overrightarrow{v}\) without hitting another element of S. We say that \(S_i \prec S_j\) if \(S_j\) has to be removed before we can remove \(S_i\). The relation \(\prec \) defines a partial order \(P(S,\prec )\) on S which we call the separability order of S and \( \overrightarrow{v}\). A partial order \(P(X, \prec ')\) on \(X=\{x_1, \ldots , x_n\}\) is called a separability order if there is a collection of convex sets S and a vector \( \overrightarrow{v}\) in some \(\mathbb {R}^d\) such that \(x_i \prec ' x_j\) in \(P(X, \prec ')\) if and only if \(S_i \prec S_j\) in \(P(S,\prec )\). We prove that every partial order is the separability order of a collection of convex sets in \(\mathbb {R}^4\), and that any poset of dimension 2 is the separability order of a set of line segments in \(\mathbb {R}^3\). We then study the case when the convex sets are restricted to be boxes in d-dimensional spaces. We prove that any partial order is the separability order of a family of disjoint boxes in \(\mathbb {R}^d\) for some \(d \le \lfloor \frac{n}{2} \rfloor +1\). We prove that every poset of dimension 3 has a subdivision that is the separability order of boxes in \(\mathbb {R}^3\), that there are partial orders of dimension 2 that cannot be realized as box separability in \(\mathbb {R}^3\) and that for any d there are posets with dimension d that are separability orders of boxes in \(\mathbb {R}^3\). We also prove that for any d there are partial orders with box separability dimension d; that is, d is the smallest dimension for which they are separable orders of sets of boxes in \(\mathbb {R}^d\).
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Acknowledgements
We thank José Luis Álvarez-Rebollar for his valuable remarks on the writing of this paper. José-Miguel Díaz-Báñez partially supported by Ministerio de Ciencia e Innovación CIN/AEI/10.13039/501100011033/ (PID2020-114154RB-I00) and European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement #734922. Paul Horn and Alex Stevens partially supported by Simons Collaboration Grant #525039. Mario A. Lopez partially supported by a University of Denver Evans Research Fund. Nestaly Marín supported by SEP-CONACYT of Mexico. Adriana Ramírez-Vigueras and Jorge Urrutia partially supported by PAPIIT IN105221, Programa de Apoyo a la Investigación e Innovación Tecnológica UNAM.
Funding
Partially supported by Ministerio de Ciencia e Innovación CIN/AEI/10.13039/501100011033/ (PID2020-114154RB-I00) and European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement #734922. Partially supported by Simons Collaboration Grant #525039. Partially supported by a University of Denver Evans Research Fund. Supported by SEP-CONACYT of Mexico. Partially supported by PAPIIT IN105221, Programa de Apoyo a la Investigación e Innovación Tecnológica UNAM.
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Díaz-Báñez, JM., Horn, P., Lopez, M.A. et al. Separability, Boxicity, and Partial Orders. Order 40, 699–712 (2023). https://doi.org/10.1007/s11083-023-09628-8
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DOI: https://doi.org/10.1007/s11083-023-09628-8