Coincidences of simplex centers and related facial structures

Pre-kites: Simplices having a regular facet

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…