New counterexamples to Knaster's conjecture

Preprint series: 03-09, Reports on Analysis and Algebra and Geometry

MSC:
52A20 Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45]
Abstract: Given a continuous map $f: \mathbb{S}^{n-1} \rightarrow \mathbb{R}^m$ and $n-m+1$ points $p_1,\ldots,p_{n-m+1} \in \mathbb{S}^{n-1}$, does there exist a rotation $\varrho \in SO(n)$ such that $f(\varrho(p_1))=\ldots=f(\varrho(p_{n-m+1}))$? We give a negative answer to this question for $m=1$ if $n \in \{61,63,65\}$ or $n \ge 67$ and for $m=2$ if $n \ge 5$.