Pre-kites: Simplices having a regular facet

TL;DR

This paper explores the geometric properties of simplices with a regular facet, called pre-kites, analyzing their centers and regularity conditions using linear algebra and determinants.

Contribution

It introduces the concept of pre-kites, investigates their regularity conditions, and extends known results about kites to higher dimensions using linear algebra tools.

Findings

Pre-kites have specific regularity properties related to their centers.

The intersection of pre-kites with special families yields kites for dimensions n ≥ 3.

A closed-form determinant generalizes previous determinants in higher-dimensional geometry.

Abstract

The investigation of the relation among the distances of an arbitrary point in the Euclidean space $\mathbb{R}^n$ to the vertices of a regular $n$-simplex in that space has led us to the study of simplices having a regular facet. Calling an $n$-simplex with a regular facet an $n$-pre-kite, we investigate, in the spirit of [4], [10], [9], and [15], and using tools from linear algebra, the degree of regularity implied by the coincidence of any two of the classical centers of such simplices. We also prove that if $n \ge 3$, then the intersection of the family of $n$-pre-kites with any of the four known special families is the family of $n$-kites, thus extending the result in [18]. A basic tool is a closed form of a determinant that arises in the context of a certain Cayley-Menger determinant, and that generalizes several determinants that appear in [9], [15], and [16]. Thus the paper is a further testimony to the special role that linear algebra plays in higher dimensional geometry.