The smallest mono-unstable, homogeneous convex polyhedron has at least 7 vertices Article Swipe

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Sándor Bozóki , G. Domokos , Dávid Papp , Krisztina Regős ·

YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2401.17906

We prove that every homogeneous convex polyhedron with only one unstable equilibrium (known as a mono-unstable convex polyhedron) has at least $7$ vertices. Although it has been long known that no mono-unstable tetrahedra exist, and mono-unstable polyhedra with as few as $18$ vertices and faces have been constructed, this is the first nontrivial lower bound on the number of vertices for a mono-unstable polyhedron. There are two main ingredients in the proof. We first establish two types of relationships, both expressible as (non-convex) quadratic inequalities, that the coordinates of the vertices of a mono-unstable convex polyhedron must satisfy, taking into account the combinatorial structure of the polyhedron. Then we use numerical semidefinite optimization algorithms to compute easily and independently verifiable, rigorous certificates that the resulting systems of quadratic inequalities (5943 in total) are indeed inconsistent in each case.

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Polyhedron Homogeneous Regular polygon Convex polytope Combinatorics Mathematics Convex analysis Geometry Convex optimization

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2401.17906
PDF: https://arxiv.org/pdf/2401.17906
OA Status: green
Related Works: 10
OpenAlex ID: https://openalex.org/W4391463031

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DOI: https://doi.org/10.48550/arxiv.2401.17906

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Title: The smallest mono-unstable, homogeneous convex polyhedron has at least 7 vertices

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Language: en

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Publication year: 2024

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Publication date: 2024-01-31

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Authors: Sándor Bozóki, G. Domokos, Dávid Papp, Krisztina Regős

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Landing page: https://arxiv.org/abs/2401.17906

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Concepts: Polyhedron, Homogeneous, Regular polygon, Convex polytope, Combinatorics, Mathematics, Convex analysis, Geometry, Convex optimization

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