Properties of distance functions on convex surfaces and applications

Jan Rataj; Luděk Zajíček

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 1, page 247-269
  • ISSN: 0011-4642

Abstract

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If X is a convex surface in a Euclidean space, then the squared intrinsic distance function dist 2 ( x , y ) is DC (d.c., delta-convex) on X × X in the only natural extrinsic sense. An analogous result holds for the squared distance function dist 2 ( x , F ) from a closed set F ⊂ X . Applications concerning r -boundaries (distance spheres) and ambiguous loci (exoskeletons) of closed subsets of a convex surface are given.

How to cite

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Rataj, Jan, and Zajíček, Luděk. "Properties of distance functions on convex surfaces and applications." Czechoslovak Mathematical Journal 61.1 (2011): 247-269. <http://eudml.org/doc/196902>.

@article{Rataj2011,
abstract = {If $X$ is a convex surface in a Euclidean space, then the squared intrinsic distance function $\mathop \{\{\rm dist\}\}^2(x,y)$ is DC (d.c., delta-convex) on $X\times X$ in the only natural extrinsic sense. An analogous result holds for the squared distance function $\mathop \{\{\rm dist\}\}^2(x,F)$ from a closed set $F \subset X$. Applications concerning $r$-boundaries (distance spheres) and ambiguous loci (exoskeletons) of closed subsets of a convex surface are given.},
author = {Rataj, Jan, Zajíček, Luděk},
journal = {Czechoslovak Mathematical Journal},
keywords = {distance function; convex surface; Alexandrov space; DC manifold; ambiguous locus; skeleton; $r$-boundary; distance function; convex surface; Alexandrov space; DC manifold; ambiguous locus; skeleton; -boundary},
language = {eng},
number = {1},
pages = {247-269},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Properties of distance functions on convex surfaces and applications},
url = {http://eudml.org/doc/196902},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Rataj, Jan
AU - Zajíček, Luděk
TI - Properties of distance functions on convex surfaces and applications
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 1
SP - 247
EP - 269
AB - If $X$ is a convex surface in a Euclidean space, then the squared intrinsic distance function $\mathop {{\rm dist}}^2(x,y)$ is DC (d.c., delta-convex) on $X\times X$ in the only natural extrinsic sense. An analogous result holds for the squared distance function $\mathop {{\rm dist}}^2(x,F)$ from a closed set $F \subset X$. Applications concerning $r$-boundaries (distance spheres) and ambiguous loci (exoskeletons) of closed subsets of a convex surface are given.
LA - eng
KW - distance function; convex surface; Alexandrov space; DC manifold; ambiguous locus; skeleton; $r$-boundary; distance function; convex surface; Alexandrov space; DC manifold; ambiguous locus; skeleton; -boundary
UR - http://eudml.org/doc/196902
ER -

References

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