The multiple points of maps from sphere to Euclidean space

In this paper, we obtain some sufficient conditions to guarantee the existence of multiple points of maps from $S^m$ to $\mathbb{R}^d$. Our main tool is the ideal-valued index of $G$-space defined by E. Fadell and S. Husseini. We obtain more detailed relative positional relationship of multiple points. It is proved that for a continuous real value function $f: S^m\rightarrow \mathbb{R}$ such that $f(-p)=-f(p)$, if $m+1$ is a power of $2$, then there are $m+1$ points $p_1, \ldots, p_{m+1}$ in $S^m$ such that $f(p_1)=\cdots=f(p_{m+1})$, where $p_1, \ldots, p_{m+1}$ are linearly dependent and any $m$ points of $p_1, \ldots, p_{m+1}$ are linearly independent. As a generalization of Hopf's theorem, we also prove that for any continuous map $f: S^m\rightarrow \mathbb{R}^d$, if $m> d$, then there exists a pair of mutually orthogonal points having the same image in addition to the antipodal points.