Mathematical Programming

, Volume 166, Issue 1–2, pp 273–295 | Cite as

Realizability and inscribability for simplicial polytopes via nonlinear optimization

  • Moritz Firsching
Full Length Paper Series A


We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. In order to show non-realizability of simplicial spheres, we extend the method of finding biquadratic final polynomials for matroid polytopes to partial matroid polytopes. Combining these two methods we obtain a complete classification of neighborly polytopes of dimension 4, 6 and 7 with 11 vertices, of neighborly 5-polytopes with 10 vertices, as well as a complete classification of simplicial 3-spheres with 10 vertices into polytopal and non-polytopal spheres. Surprisingly many of the realizable polytopes are also inscribable.

Mathematics Subject Classification

52B11 52C40 52B40 90C30 



I am indebted to Arnau Padrol, who started the project by telling me about the problem discussed in Sect. 3.4 and proposing to use optimization methods in order to solve it and to Hao Chen, who got very interested in proving non-inscribability results. I am thankful to Hiroyuki Miyata for the help while working on Sect. 3.1; Hiroyuki Miyata was kind enough to run the biquadratic final polynomial method for us until Arnau Padrol and I had implemented it and to run double checks afterwards. For the help in Sect. 3.2 I thank Frank Lutz, who provided the list of triangulations and David Bremner, who decided the existence of a compatible matroid polytope for these triangulations. I enjoyed discussions on the topic discussed in Sect. 3.3 with Florian Frick, John Sullivan and Frank Lutz, who again provided a list of triangulations. I would like to thank Moritz W. Schmitt, Philip Brinkmann and Louis Theran for valuable discussions and Benjamin Müller as well as Ambros Gleixner for help with my questions regarding SCIP. I thank two anonymous referees for their constructive suggestions for improvements. Last but not least, I am very grateful to Günter M. Ziegler for helping at every step in the preparation of this paper.

Funding was provided by Deutsche Forschungsgemeinschaft (Grant No. SFB/TRR109).


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Authors and Affiliations

  1. 1.Institut für MathematikFreie Universität BerlinBerlinGermany

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