Abstract
In this paper we use geometric dissection to obtain linear equations on the flag vectors on convex polytopes. These results provide new proofs and expressions of the complete system of such equations originally discovered by Bayer and Billera. The Mayer-Vietoris equation applies to a situation where two convex polytopes overlap to produce union and intersection, both convex polytopes. The operatorsI andC applied to a polytope produce the cylinder (or prism) and cone (or pyramid), respectively, with the given polytopes as base. TheIC equation relates the flag vectors of the polytopes obtained in this way. As a consequence, it becomes easier to define linear functios of the flag vector, via initial data and their law of transformation under the operatorsI andC.
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Fine, J. The Mayer-Vietoris andIC equations for convex polytopes. Discrete Comput Geom 13, 177–188 (1995). https://doi.org/10.1007/BF02574036
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DOI: https://doi.org/10.1007/BF02574036