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

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 $\mathbb{R}^3$.