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

Abstract

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. Blagojević and G. Ziegler \cite{blagojevictetrahedra} where Fadell-Husseini index is used for solving a problem related to the topological Borsuk problem for ${\mathbb{R}}^3$.