Order Types of convex bodies

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.