On convex cones with infinitely many critical angles

This note deals with some cardinality issues concerning the set of critical angles of a convex cone . Such set is referred to as the angular spectrum of the cone. In a recent work of ours, it has been shown that the angular spectrum of a polyhedral cone is necessarily finite and that its cardinality...

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Bibliographic Details
Published inOptimization Vol. 56; no. 1-2; pp. 115 - 128
Main Authors Iusem, Alfredo, Seeger, Alberto
Format Journal Article
LanguageEnglish
Published Philadelphia Taylor & Francis Group 01.02.2007
Taylor & Francis LLC
Taylor & Francis
Subjects
Angular spectra
Cantor ternary set
Convex cones
Critical angles
Mathematical problems
Mathematics
Mathematics Subject Classifications 2000
Polyhedra
Polynomials
Studies
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Summary:This note deals with some cardinality issues concerning the set of critical angles of a convex cone . Such set is referred to as the angular spectrum of the cone. In a recent work of ours, it has been shown that the angular spectrum of a polyhedral cone is necessarily finite and that its cardinality can grow at most polynomially with respect to the number of generators. In this note, we explore the case of nonpolyhedral cones. More specifically, we construct a cone whose angular spectrum is infinite (but possibly countable), and, what is harder to achieve, we construct a cone with noncountable angular spectrum. The construction procedure is highly technical in both cases, but the obtained results are useful for better understanding why some convex cones exhibit such a complicated angular structure.
Bibliography:SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ISSN:0233-1934
1029-4945
DOI:10.1080/02331930600819985