It is known that for every continuous map $M:S^1\to \mathbb{R}$ there are infinite double points.also by borsuk-ulam theorem this is true for each map $N:S^n\to \mathbb{R}^n, n\in \mathbb{N}$. A question arising here concerns finding k-manifold $X$ which each map $S:X \to \mathbb{R}^k$ has a triple point.
A: Does there exist such $X$? if yes then how it would be figure out?
B: Does there exist a minimal $X$? (means the number of triple points of a map $S:X \to \mathbb{R}^k$ be minimum among all $S$ and all $X$.)
Note: a function $f:X\to Y$ have a double/triple point if for a point $p\in Y$ the preimage $f^{-1}(p)$ consist of at least 2/3 points in $X$.