Continuity of the Visibility Function in the Boundary

The visibility function of a compact set S ⊂ Ed assigns to each x ∈ S the Lebesgue outer measure of its star in S. This function was introduced by G. Beer in 1972. In 1991, A. Forte Cunto characterized the points of discontinuity of the visibility function in the boundary of a planar Jordan domain....

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Autores principales: Piacquadio Losada, M., Forte Cunto, A., Toranzos, F.A.
Formato: JOUR
Materias:
Beer's visibility function
Point of restricted visibility
Starshaped set
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00465755_v80_n1-3_p43_PiacquadioLosada
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id todo:paper_00465755_v80_n1-3_p43_PiacquadioLosada
record_format dspace
spelling todo:paper_00465755_v80_n1-3_p43_PiacquadioLosada2023-10-03T14:52:17Z Continuity of the Visibility Function in the Boundary Piacquadio Losada, M. Forte Cunto, A. Toranzos, F.A. Beer's visibility function Point of restricted visibility Starshaped set The visibility function of a compact set S ⊂ Ed assigns to each x ∈ S the Lebesgue outer measure of its star in S. This function was introduced by G. Beer in 1972. In 1991, A. Forte Cunto characterized the points of discontinuity of the visibility function in the boundary of a planar Jordan domain. In a recent paper, the present authors extended this characterization to compact subsets of Ed with certain topological restrictions. These restrictions are removed here and it is proved that the visibility function of a compact subset of Ed is continuous at a point if and only if the set of restricted visibility of this point has null Lebesgue outer measure. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00465755_v80_n1-3_p43_PiacquadioLosada
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Beer's visibility function
Point of restricted visibility
Starshaped set
spellingShingle Beer's visibility function
Point of restricted visibility
Starshaped set
Piacquadio Losada, M.
Forte Cunto, A.
Toranzos, F.A.
Continuity of the Visibility Function in the Boundary
topic_facet Beer's visibility function
Point of restricted visibility
Starshaped set
description The visibility function of a compact set S ⊂ Ed assigns to each x ∈ S the Lebesgue outer measure of its star in S. This function was introduced by G. Beer in 1972. In 1991, A. Forte Cunto characterized the points of discontinuity of the visibility function in the boundary of a planar Jordan domain. In a recent paper, the present authors extended this characterization to compact subsets of Ed with certain topological restrictions. These restrictions are removed here and it is proved that the visibility function of a compact subset of Ed is continuous at a point if and only if the set of restricted visibility of this point has null Lebesgue outer measure.
format JOUR
author Piacquadio Losada, M.
Forte Cunto, A.
Toranzos, F.A.
author_facet Piacquadio Losada, M.
Forte Cunto, A.
Toranzos, F.A.
author_sort Piacquadio Losada, M.
title Continuity of the Visibility Function in the Boundary
title_short Continuity of the Visibility Function in the Boundary
title_full Continuity of the Visibility Function in the Boundary
title_fullStr Continuity of the Visibility Function in the Boundary
title_full_unstemmed Continuity of the Visibility Function in the Boundary
title_sort continuity of the visibility function in the boundary
url http://hdl.handle.net/20.500.12110/paper_00465755_v80_n1-3_p43_PiacquadioLosada
work_keys_str_mv AT piacquadiolosadam continuityofthevisibilityfunctionintheboundary
AT fortecuntoa continuityofthevisibilityfunctionintheboundary
AT toranzosfa continuityofthevisibilityfunctionintheboundary
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