On fiber diameters of continuous maps

Peter S. Landweber; Emanuel A. Lazar; Neel Patel

doi:10.4169/amer.math.monthly.123.4.392

On fiber diameters of continuous maps

Peter S. Landweber, Emanuel A. Lazar, Neel Patel

Research output: Contribution to journal › Article › peer-review

Abstract

We present a surprisingly short proof that for any continuous map f: ℝⁿ → 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.

Original language	English
Pages (from-to)	392-397
Number of pages	6
Journal	American Mathematical Monthly
Volume	123
Issue number	4
DOIs	https://doi.org/10.4169/amer.math.monthly.123.4.392
State	Published - 2016
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics

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10.4169/amer.math.monthly.123.4.392

Cite this

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title = "On fiber diameters of continuous maps",

abstract = "We present a surprisingly short proof that for any continuous map f: ℝn → ℝm, 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.",

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AU - Lazar, Emanuel A.

AU - Patel, Neel

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PY - 2016

Y1 - 2016

N2 - We present a surprisingly short proof that for any continuous map f: ℝn → ℝm, 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.

AB - We present a surprisingly short proof that for any continuous map f: ℝn → ℝm, 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.

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DO - 10.4169/amer.math.monthly.123.4.392

M3 - مقالة

SN - 0002-9890

VL - 123

SP - 392

EP - 397

JO - American Mathematical Monthly

JF - American Mathematical Monthly

IS - 4

ER -

On fiber diameters of continuous maps

Abstract

All Science Journal Classification (ASJC) codes

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