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The smallest mono-unstable convex polyhedron with point masses has 8 faces and 11 vertices

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  • Papp, Dávid
  • Regős, Krisztina
  • Domokos, Gábor
  • Bozóki, Sándor

Abstract

In the study of monostatic polyhedra, initiated by John H. Conway in 1966, the main question is to construct such an object with the minimal number of faces and vertices. By distinguishing between various material distributions and stability types, this expands into a small family of related questions. While many upper and lower bounds on the necessary numbers of faces and vertices have been established, none of these questions has been so far resolved. Adapting an algorithm presented in Bozóki et al. (2022), here we offer the first complete answer to a question from this family: by using the toolbox of semidefinite optimization to efficiently generate the hundreds of thousands of infeasibility certificates, we provide the first-ever proof for the existence of a monostatic polyhedron with point masses, having minimal number (V=11) of vertices (Theorem 3) and a minimal number (F=8) of faces. We also show that V=11 is the smallest number of vertices that a mono-unstable polyhedron can have in all dimensions greater than 1 (Corollary 6).

Suggested Citation

  • Papp, Dávid & Regős, Krisztina & Domokos, Gábor & Bozóki, Sándor, 2023. "The smallest mono-unstable convex polyhedron with point masses has 8 faces and 11 vertices," European Journal of Operational Research, Elsevier, vol. 310(2), pages 511-517.
    • Handle: RePEc:eee:ejores:v:310:y:2023:i:2:p:511-517
    DOI: 10.1016/j.ejor.2023.04.028
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    References listed on IDEAS

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    1. Campos, Juan S. & Misener, Ruth & Parpas, Panos, 2019. "A multilevel analysis of the Lasserre hierarchy," European Journal of Operational Research, Elsevier, vol. 277(1), pages 32-41.
    2. Klerk, Etienne de, 2010. "Exploiting special structure in semidefinite programming: A survey of theory and applications," European Journal of Operational Research, Elsevier, vol. 201(1), pages 1-10, February.
    3. Lai, Xiangjing & Hao, Jin-Kao & Yue, Dong & Lü, Zhipeng & Fu, Zhang-Hua, 2022. "Iterated dynamic thresholding search for packing equal circles into a circular container," European Journal of Operational Research, Elsevier, vol. 299(1), pages 137-153.
    4. Immanuel M. Bomze & Werner Schachinger & Reinhard Ullrich, 2018. "The Complexity of Simple Models—A Study of Worst and Typical Hard Cases for the Standard Quadratic Optimization Problem," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 651-674, May.
    5. Kurpel, Deidson Vitorio & Scarpin, Cassius Tadeu & Pécora Junior, José Eduardo & Schenekemberg, Cleder Marcos & Coelho, Leandro C., 2020. "The exact solutions of several types of container loading problems," European Journal of Operational Research, Elsevier, vol. 284(1), pages 87-107.
    6. de Klerk, E. & Maharry, J. & Pasechnik, D.V. & Richter, B. & Salazar, G., 2006. "Improved bounds for the crossing numbers of Km,n and Kn," Other publications TiSEM eca87811-247d-489f-89c2-c, Tilburg University, School of Economics and Management.
    7. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
    8. López, C.O. & Beasley, J.E., 2011. "A heuristic for the circle packing problem with a variety of containers," European Journal of Operational Research, Elsevier, vol. 214(3), pages 512-525, November.
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