  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.
