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Algebraic and metric structures of the geometric mean cone
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Periodical
- Title:
- Special matrices
- Publication:
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Warsaw: De Gruyter, Versita
- Note:
- Gesehen am 23.01.14
- Open Access
- Namensnennung 4.0 International
- Scope:
- Online-Ressource
- ISSN:
- 2300-7451
- ZDB-ID:
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2753755-9
- VÖBB-Katalog:
- 35278167
- Keywords:
- Zeitschrift
- Classification:
- Mathematik
- Collection:
- Mathematik
- Copyright:
- Rights reserved
- Accessibility:
- Free Access
- Title:
- Special matrices
- Publication:
-
Warsaw: De Gruyter, Versita
- Note:
- Gesehen am 23.01.14
- Open Access
- Namensnennung 4.0 International
- Scope:
- Online-Ressource
- ISSN:
- 2300-7451
- ZDB-ID:
-
2753755-9
- VÖBB-Katalog:
- 35278167
- Keywords:
- Zeitschrift
- Classification:
- Mathematik
- Collection:
- Mathematik
- Copyright:
- Rights reserved
- Accessibility:
- Free Access
Article
- Title:
- Algebraic and metric structures of the geometric mean cone
- Publication:
-
Warsaw: De Gruyter, Versita, 2026
- Language:
- English
- Information:
- Abstract: The geometric mean cone (GMC) is a higher-order generalization of the second-order cone and the rotated quadratic cone. While these lower-order cones admit well-established Euclidean Jordan algebraic representations, no analogous framework is currently known for GMCs of order three or higher. In this paper, we study the algebraic structure of the GMC by extending the algebra of the rotated quadratic cone to higher orders. Specifically, we introduce and analyze notions such as a bilinear product, linear matrix representation, characteristic polynomial, rank, and a trace inner product. By adopting the trace inner product, we show that, contrary to certain claims in the existing literature, the GMC is self-dual. While the algebras of both the rotated quadratic cone and the second-order cone are Euclidean Jordan algebras, we show that the extended algebra of the GMC is a Euclidean metrized algebra. However, for orders three and higher, no quadratic representation exists in the extended algebra that maps the entire ambient space into the GMC. Consequently, it remains an open question whether the power cone admits a quadratic representation. Our results clarify the precise sense in which the GMC admits an algebraic structure, with potential implications for logarithmic barrier constructions in interior-point methods over GMCs.
- Scope:
- Online-Ressource
- Note:
- Open Access
- Archivierung/Langzeitarchivierung gewährleistet
- Keywords:
- Jordan algebras ; metrized algebras ; bilinear forms ; convex cones ; 15A63 ; 52A41 ; 90C25 ; 47L07
- Classification:
- Mathematik
- Collection:
- Mathematik
- Copyright:
- CC BY
- Accessibility:
- Free Access
- Title:
- Algebraic and metric structures of the geometric mean cone
- Publication:
-
Warsaw: De Gruyter, Versita, 2026
- Language:
- English
- Information:
- Abstract: The geometric mean cone (GMC) is a higher-order generalization of the second-order cone and the rotated quadratic cone. While these lower-order cones admit well-established Euclidean Jordan algebraic representations, no analogous framework is currently known for GMCs of order three or higher. In this paper, we study the algebraic structure of the GMC by extending the algebra of the rotated quadratic cone to higher orders. Specifically, we introduce and analyze notions such as a bilinear product, linear matrix representation, characteristic polynomial, rank, and a trace inner product. By adopting the trace inner product, we show that, contrary to certain claims in the existing literature, the GMC is self-dual. While the algebras of both the rotated quadratic cone and the second-order cone are Euclidean Jordan algebras, we show that the extended algebra of the GMC is a Euclidean metrized algebra. However, for orders three and higher, no quadratic representation exists in the extended algebra that maps the entire ambient space into the GMC. Consequently, it remains an open question whether the power cone admits a quadratic representation. Our results clarify the precise sense in which the GMC admits an algebraic structure, with potential implications for logarithmic barrier constructions in interior-point methods over GMCs.
- Scope:
- Online-Ressource
- Note:
- Open Access
- Archivierung/Langzeitarchivierung gewährleistet
- Keywords:
- Jordan algebras ; metrized algebras ; bilinear forms ; convex cones ; 15A63 ; 52A41 ; 90C25 ; 47L07
- Classification:
- Mathematik
- Collection:
- Mathematik
- Copyright:
- CC BY
- Accessibility:
- Free Access
