Spectral Polyhedra
A spectral convex set is a collection of symmetric matrices whose range of eigenvalues forms a symmetric convex set. Spectral convex sets generalize the Schur-Horn orbitopes studied by Sanyal–Sottile–Sturmfels (2011). We study this class of convex bodies, which is closed under intersections, polarit...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000628/type/journal_article |
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| author | Raman Sanyal James Saunderson |
| author_facet | Raman Sanyal James Saunderson |
| author_sort | Raman Sanyal |
| collection | DOAJ |
| description | A spectral convex set is a collection of symmetric matrices whose range of eigenvalues forms a symmetric convex set. Spectral convex sets generalize the Schur-Horn orbitopes studied by Sanyal–Sottile–Sturmfels (2011). We study this class of convex bodies, which is closed under intersections, polarity and Minkowski sums. We describe orbits of faces and give a formula for their Steiner polynomials. We then focus on spectral polyhedra. We prove that spectral polyhedra are spectrahedra and give small representations as spectrahedral shadows. We close with observations and questions regarding hyperbolicity cones, polar convex bodies and spectral zonotopes. |
| format | Article |
| id | doaj-art-c55859b2b6d5442283b2bba6fc8836ca |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-c55859b2b6d5442283b2bba6fc8836ca2025-02-06T09:14:58ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.62Spectral PolyhedraRaman Sanyal0https://orcid.org/0000-0001-9127-7272James Saunderson1https://orcid.org/0000-0002-5456-0180Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, Frankfurt am Main, D-60325, Germany; E-mail:Department of Electrical and Computer Systems Engineering, Monash University, Clayton, VIC 3800, AustraliaA spectral convex set is a collection of symmetric matrices whose range of eigenvalues forms a symmetric convex set. Spectral convex sets generalize the Schur-Horn orbitopes studied by Sanyal–Sottile–Sturmfels (2011). We study this class of convex bodies, which is closed under intersections, polarity and Minkowski sums. We describe orbits of faces and give a formula for their Steiner polynomials. We then focus on spectral polyhedra. We prove that spectral polyhedra are spectrahedra and give small representations as spectrahedral shadows. We close with observations and questions regarding hyperbolicity cones, polar convex bodies and spectral zonotopes.https://www.cambridge.org/core/product/identifier/S2050509424000628/type/journal_article52A0590C2252A4152A3952B12 |
| spellingShingle | Raman Sanyal James Saunderson Spectral Polyhedra Forum of Mathematics, Sigma 52A05 90C22 52A41 52A39 52B12 |
| title | Spectral Polyhedra |
| title_full | Spectral Polyhedra |
| title_fullStr | Spectral Polyhedra |
| title_full_unstemmed | Spectral Polyhedra |
| title_short | Spectral Polyhedra |
| title_sort | spectral polyhedra |
| topic | 52A05 90C22 52A41 52A39 52B12 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000628/type/journal_article |
| work_keys_str_mv | AT ramansanyal spectralpolyhedra AT jamessaunderson spectralpolyhedra |