  We introduce the concept of $r$-equilateral $m$-gons. We prove the existence
of $r$-equilateral $p$-gons in $\mathbb R^d$ if $r<d$ and the existence of
equilateral $p$-gons in the image of continuous injective maps $f:S^d\to
\mathbb R^{d+1}$. Our ideas are based mainly in the paper of Y. Soibelman
\cite{soibelman}, in which the topological Borsuk number of $\mathbb{R}^2$ is
calculated by means of topological methods and the paper of P. Blagojevi\'c and
G. Ziegler \cite{blagojevictetrahedra} where Fadell-Husseini index is used for
solving a problem related to the topological Borsuk problem for $\mathbb{R}^3$.
