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We introduce the concept of $r$-equilateral $m$-gons. We prove the existenceof $r$-equilateral $p$-gons in $\mathbb R^d$ if $r<d$ and the existence ofequilateral $p$-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 $\mathbb{R}^2$ iscalculated by means of topological methods and the paper of P. Blagojevi\'c andG. Ziegler \cite{blagojevictetrahedra} where Fadell-Husseini index is used forsolving a problem related to the topological Borsuk problem for $\mathbb{R}^3$.

Equilateral $p$-gons in $\mathbb R^d$ and deformed spheres and mod $p$ Fadell-Husseini index