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On the concept of center for geometric objects and related problems

M. Magdalena Martínez-Rico, L. Felipe Prieto-Martínez, R. Sánchez-Cauce

TL;DR

The paper proposes a unified framework in which centers of geometric objects are modeled as equivariant maps under the similarity group, enabling a single theory to encompass diverse spaces such as multisets, polygons, and Borel sets. It elucidates Kimberling-style barycentric representations for triangle centers and characterizes when centers admit simple, coordinate-like expressions, linking these forms to symmetry and continuity considerations. It also presents an axiom-of-choice construction that guarantees the existence of centers with prescribed fixed points, while highlighting open questions about continuity and coincidence across spaces. Beyond centers, the work discusses central objects like lines and automorphisms that share the same equivariance principle, illustrating a broader landscape of translation-invariant geometric constructions with potential applications in imaging, geography, and beyond.

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.

On the concept of center for geometric objects and related problems

TL;DR

The paper proposes a unified framework in which centers of geometric objects are modeled as equivariant maps under the similarity group, enabling a single theory to encompass diverse spaces such as multisets, polygons, and Borel sets. It elucidates Kimberling-style barycentric representations for triangle centers and characterizes when centers admit simple, coordinate-like expressions, linking these forms to symmetry and continuity considerations. It also presents an axiom-of-choice construction that guarantees the existence of centers with prescribed fixed points, while highlighting open questions about continuity and coincidence across spaces. Beyond centers, the work discusses central objects like lines and automorphisms that share the same equivariance principle, illustrating a broader landscape of translation-invariant geometric constructions with potential applications in imaging, geography, and beyond.

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.
Paper Structure (9 sections, 24 equations, 1 figure)

This paper contains 9 sections, 24 equations, 1 figure.

Table of Contents

  1. Introduction.
  2. Some examples.
  3. Centers of multisets.
  4. Centers of polygons.
  5. Centers of Borel sets.
  6. Centers with other type of spaces of geometric objects.
  7. First Question.
  8. Second Question.
  9. Central Objects

Figures (1)

  • Figure 1: A well known method for computing centroids is the one of geometric decomposition. The centroid of the interior region (as a center of a Borel set) coincides with the centroid of the points represented with ($\bullet$) (as a center of a multiset). The centroid of the border (as a centroid of a Borel set) coincides with the centroid of the points represented with "$\times$" (as a centroid of the multiset). These centroids are marked with a triangle ($\blacktriangle$) and a square ($\blacksquare$), respectively, and do not coincide.