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

**MSC:**- 55M20 Fixed points and coincidences [See also 54H25]
- 52A20 Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45]
- 54H25 Fixed-point and coincidence theorems [See also 47H10, 55M20]

**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 $n \ge 5$.

**Keywords:** *Knaster's conjecture, Borsuk-Ulam theorem, continuous functions on spheres, level sets of supremum norms*

**Upload:** 2003
-07
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**Update:** 2003
-07
-01

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