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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.03698 |
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| _version_ | 1866915539902791680 |
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| author | Avis, David |
| author_facet | Avis, David |
| contents | A polyhedron is pointed if it contains at least one vertex. Every pointed polyhedron P in R^n can be described by an H-representation consisting of half spaces or equivalently by a V-representation consisting of the convex hull of a set of vertices and extreme rays. We can define matrices H(P) and V(P), each with n + 1 columns, that encode these representations. Define polyhedron Q by setting H(Q)=V(P). We show that Q is the polar of P. Call P HV-symmetric if V(Q) in turn encodes the H-representation of P. It is well known and often stated that polytopes that contain the origin in their interior and pointed polyhedral cones are HV-symmetric. We show here that, more generally, a pointed polyhedron with pointed polar is HV-symmetric if and only if it contains the origin. We show this using Minkowski's bipolar equation and discuss implications for the vertex and facet enumeration problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03698 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | HV-symmetric polyhedra and bipolarity Avis, David Optimization and Control 51M20 A polyhedron is pointed if it contains at least one vertex. Every pointed polyhedron P in R^n can be described by an H-representation consisting of half spaces or equivalently by a V-representation consisting of the convex hull of a set of vertices and extreme rays. We can define matrices H(P) and V(P), each with n + 1 columns, that encode these representations. Define polyhedron Q by setting H(Q)=V(P). We show that Q is the polar of P. Call P HV-symmetric if V(Q) in turn encodes the H-representation of P. It is well known and often stated that polytopes that contain the origin in their interior and pointed polyhedral cones are HV-symmetric. We show here that, more generally, a pointed polyhedron with pointed polar is HV-symmetric if and only if it contains the origin. We show this using Minkowski's bipolar equation and discuss implications for the vertex and facet enumeration problems. |
| title | HV-symmetric polyhedra and bipolarity |
| topic | Optimization and Control 51M20 |
| url | https://arxiv.org/abs/2406.03698 |