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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| Published in | Optimization Vol. 56; no. 1-2; pp. 115 - 128 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Philadelphia
Taylor & Francis Group
01.02.2007
Taylor & Francis LLC Taylor & Francis |
| Subjects | |
| Online Access | Get full text |
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| Abstract | 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. |
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| AbstractList | 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. [PUBLICATION ABSTRACT] 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. |
| Author | Seeger, Alberto Iusem, Alfredo |
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| Copyright | Copyright Taylor & Francis Group, LLC 2007 Copyright Taylor & Francis Group Feb-Apr 2007 Distributed under a Creative Commons Attribution 4.0 International License |
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| References | Willard S (CIT0011) 1970 Iusem A (CIT0004) 2006 Iusem A (CIT0002) 2005; 24 Dujundji J (CIT0001) 1966 Wets RJB (CIT0010) 1980 CIT0007 Iusem A (CIT0003) 2005; 104 Quittner P (CIT0006) 1986; 27 Peña J (CIT0005) 2000; 87 CIT0009 CIT0008 |
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| Snippet | 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... 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... |
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| SubjectTerms | Angular spectra Cantor ternary set Convex cones Critical angles Mathematical problems Mathematics Mathematics Subject Classifications 2000 Polyhedra Polynomials Studies |
| Title | On convex cones with infinitely many critical angles |
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