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

TL;DR

This paper investigates inequalities relating the counts of convex polytopes in d-dimensional space that either contain or avoid zero, based on their vertices being subsets of a fixed finite point set.

Contribution

It establishes new inequalities between the numbers of convex polytopes containing or avoiding zero with vertices from a specified finite set.

Findings

Derived inequalities between convex polytopes containing and avoiding zero

Provided bounds based on the structure of the finite point set

Enhanced understanding of convex polytope enumeration in relation to zero containment

Abstract

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.