Functions defined on spheres -- remarks on a paper by K. Zarankiewicz (Q2770174)

Let \(S^n\) be an \(n\)-sphere. It is shown that for any two mappings \(\varphi: S^n\to S^n\) and \(f:S^n\to Y\), where \(n\geq 2\) and \(Y\) is 1-dimensional, there are a point \(x_0\in S^n\) and a continuum \(C\subset S^n\) connecting \(x_0\) with \(\varphi (x_0)\) such that \(f\) is constant on \(C\). This generalizes a result of \textit{L. M. Sonneborn} [Pac. J. Math. 13, 297-303 (1963; Zbl 0113.37904)] (who proved this for \(n=2\) with \(Y\) being the real line \(\mathbb{R})\) and answers a question of \textit{A. Lelek} [Wiadomości Matematyczne 4, 296 (1991)]. If (still for \(n=2\) and \(Y=\mathbb{R})\) \(f\) is equivariant with respect to the canonical involutions, then \(C\) can be chosen to be symmetric. This improves a classical result of \textit{G. R. Livesay} [Ann. Math. (2) 59, 227-229 (1954; Zbl 0056.41901)] and \textit{K. Zarankiewicz} [Bull. Acad. Polon. Sci., Cl. III 2, 117-120 (1954; Zbl 0056.41902)]. Further, among other results, it is also proved that for any \(n\geq 2\) and for any mapping \(f:S^n\to \mathbb{R}\) the Borsuk-Ulam set \(A(f)=\{x \in S^n: f(x)=f(-x)\}\) contains a unique symmetric component \(D\); it separates \(S^n\) between antipodal points \(x\) and \(-x\) for each \(x\in S^n\smallsetminus D\).