Order Types of convex bodies
Alfredo Hubard, Luis Montejano, Emiliano Mora, Andrew Suk

TL;DR
This paper extends the concept of order types from points to convex bodies, providing new bounds on Erdős–Szekeres theorems for convex bodies and a combinatorial characterization of their convex position.
Contribution
It introduces a combinatorial characterization for convex position of convex bodies, extending order types beyond points, and derives new bounds for related theorems.
Findings
Extended order type concept to convex bodies
Derived new bounds on Erdős–Szekeres theorems for convex bodies
Provided a combinatorial characterization of convex position
Abstract
We give new bounds on the Erdos-Szekeres theorems for convex bodies of Bisztriczky and Fejes Toth and of Pach and Toth. We derive them from a combinatorial characterization of convex position of a family of planar convex bodies. This characterization confirms that the concept of Order Type for points can be extended to noncrossing families of convex bodies in a geometrically meaningful way.
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Taxonomy
TopicsPoint processes and geometric inequalities · Computational Geometry and Mesh Generation · Limits and Structures in Graph Theory
