Remarks on WDC sets

Dušan Pokorný; Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (2021)

  • Issue: 1, page 81-94
  • ISSN: 0010-2628

Abstract

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We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets A ⊂ ℝ 2 . We prove that, for such A , the distance function d A = dist ( · , A ) is a “DC aura” for A , which implies that each closed locally WDC set in ℝ 2 is a WDC set. Another consequence is that compact WDC subsets of ℝ 2 form a Borel subset of the space of all compact sets.

How to cite

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Pokorný, Dušan, and Zajíček, Luděk. "Remarks on WDC sets." Commentationes Mathematicae Universitatis Carolinae (2021): 81-94. <http://eudml.org/doc/298283>.

@article{Pokorný2021,
abstract = {We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets $A\subset \{\mathbb \{R\}\}^2$. We prove that, for such $A$, the distance function $d_A= \{\rm dist\}(\cdot ,A)$ is a “DC aura” for $A$, which implies that each closed locally WDC set in $\{\mathbb \{R\}\}^2$ is a WDC set. Another consequence is that compact WDC subsets of $\{\mathbb \{R\}\}^2$ form a Borel subset of the space of all compact sets.},
author = {Pokorný, Dušan, Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {distance function; WDC set; DC function; DC aura; Borel complexity},
language = {eng},
number = {1},
pages = {81-94},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Remarks on WDC sets},
url = {http://eudml.org/doc/298283},
year = {2021},
}

TY - JOUR
AU - Pokorný, Dušan
AU - Zajíček, Luděk
TI - Remarks on WDC sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
IS - 1
SP - 81
EP - 94
AB - We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets $A\subset {\mathbb {R}}^2$. We prove that, for such $A$, the distance function $d_A= {\rm dist}(\cdot ,A)$ is a “DC aura” for $A$, which implies that each closed locally WDC set in ${\mathbb {R}}^2$ is a WDC set. Another consequence is that compact WDC subsets of ${\mathbb {R}}^2$ form a Borel subset of the space of all compact sets.
LA - eng
KW - distance function; WDC set; DC function; DC aura; Borel complexity
UR - http://eudml.org/doc/298283
ER -

References

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