On Convex Polytopes in the d-dimensional Space Containing and Avoiding Zero

The goal of this paper is to establish certain inequalities between the numbers of convex polytopes in the d-dimensional space “containing” and “avoiding” zero provided that their vertex sets are subsets of a given finite set of points in the space.

💡 Research Summary

The paper investigates a combinatorial geometry problem concerning convex polytopes in (\mathbb{R}^d) whose vertices are drawn from a fixed finite set (V). The central question is how many such polytopes contain the origin (denoted “containing”) versus how many avoid it (denoted “avoiding”). After establishing precise definitions—(\mathcal{C}+) for polytopes that contain the origin either in their interior or on a facet, (\mathcal{C}-) for those that do not contain the origin at all, and (\mathcal{C}_0) for polytopes that have the origin as a vertex—the author develops two main technical tools.

The first tool is a vertex‑exchange operation. Given a polytope (P\in\mathcal{C}+) with the origin in its interior, one can replace a suitably chosen vertex (v\in V) by the origin, obtaining a new polytope (P’). This operation preserves convexity and maps (\mathcal{C}+) into (\mathcal{C}-\cup\mathcal{C}0). Conversely, starting from a polytope in (\mathcal{C}-) one can sometimes insert the origin and delete a vertex, producing a member of (\mathcal{C}+). The asymmetry of this mapping yields the fundamental inequality
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