On fiber diameters of continuous maps

Abstract

We present a surprisingly short proof that for any continuous map $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$, if $n>m$, then there exists no bound on the diameter of fibers of $f$. Moreover, we show that when $m=1$, the union of small fibers of $f$ is bounded; when $m>1$, the union of small fibers need not be bounded. Applications to data analysis are considered.

6 pages, 2 figures

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• Metaphors in systolic geometry

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