On fiber diameters of continuous maps
Abstract
We present a surprisingly short proof that for any continuous map $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$, 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2015
 DOI:
 10.48550/arXiv.1503.07597
 arXiv:
 arXiv:1503.07597
 Bibcode:
 2015arXiv150307597L
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  Classical Analysis and ODEs
 EPrint:
 6 pages, 2 figures