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# Two geometric representation theorems for separoids

Periodica Mathematica Hungarica
Authors: Javier Bracho and Ricardo Strausz

## Summary

{\it Separoids\/} capture the combinatorial structure which arises from the separations by hyperplanes of a family of convex sets in some Euclidian space. Furthermore, as we prove in this note, every abstract separoid \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $S$ \end{document} can be represented by a family of convex sets in the \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(|S|-1)$ \end{document}-dimensional Euclidian space. The {\it geometric dimension\/} of the separoid is the minimum dimension where it can be represented and the upper bound given here is tight. Separoids have also the notions of {\it combinatorial dimension\/} and {\it general position\/} which are purely combinatorial in nature. In this note we also prove that: {\it a separoid in general position can be represented by a family of points if and only if its geometric and combinatorial dimensions coincide\/}.

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