The concept of center as an equivariant map and a proof of an analogue of the center conjecture for equifacetal simplices

Abstract

Several authors have remarked the convenience of understanding the different notions of center appearing in Geometry (centroid of a set of points, incenter of a triangle, center of a conic and many others) as functions. The most general way to do so is to define centers as equivariant maps between $G$-spaces. In this paper, we prove that, under certain hypothesis, for any two $G$-spaces $\mathcal{A},\mathcal{X}$, for every $V\in \mathcal{A}$ and for every point $P\in \mathcal{X}$ fixed by the symmetry group of $V$, there exists some equivariant map $\mathfrak{Z}:\mathcal{A}\to \mathcal{X}$ such that $\mathfrak{Z}(V)=P$. As a consequence of this fact, we prove an analogue (for non-neccessarily continuous centers) of the \emph{center conjecture for equifacetal simplices}, proposed by A. L. Edmonds.