On the concept of center for geometric objects and related problems

TL;DR

This paper reviews a unified, general approach to defining the center of geometric objects as an equivariant map, illustrating its broad applicability and discussing specific characterizations and open questions.

Contribution

It introduces a unifying framework for centers of geometric objects, generalizing previous approaches and exploring their properties and open problems.

Findings

The general approach encompasses many geometric spaces.

Characterizations of centers are provided for specific spaces.

Open questions are posed for future research.

Abstract

In this work, we review the concept of center of a geometric object as an equivariant map, unifying and generalizing different approaches followed by authors such as C. Kimberling or A. Edmonds. We provide examples to illustrate that this general approach encompasses many interesting spaces of geometric objects arising from different settings. Additionally, we discuss two results that characterize centers for some particular spaces of geometric objects, and we pose five open questions related to the generalization of these characterizations to other spaces. Finally, we conclude this article by briefly discussing other central objects and their relation to this concept of center.