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A convexity theorem for real projective structures

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Abstract

Given a finite collection \(\mathcal {P}\) of convex n-polytopes in \(\mathbb {R}\hbox {P}^n\) (\(n\ge 2\)), we consider a real projective manifold M which is obtained by gluing together the polytopes in \(\mathcal {P}\) along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on M is (1) convex if \(\mathcal {P}\) contains no triangular polytope, and (2) properly convex if, in addition, \(\mathcal {P}\) contains a polytope whose dual polytope is thick. Triangular polytopes and polytopes with thick duals are defined as analogues of triangles and polygons with at least five edges, respectively.

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Notes

  1. Our definition of star seems to be somewhat non-standard. We borrowed the term “residue” from [10], where residues are defined in the same way as in the present paper.

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Acknowledgments

My advisor Misha Kapovich recommended me to investigate the property of residual convexity. I am grateful to him for this and I deeply appreciate his encouragement and patience during my work. I also thank Yves Benoist and Damian Osajda for helpful discussions. The present work was partially supported by the NSF Grants DMS-04-05180 and DMS-05-54349.

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  1. School of Mathematics, KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Republic of Korea

    Jaejeong Lee

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Lee, J. A convexity theorem for real projective structures. Geom Dedicata 182, 1–41 (2016). https://doi.org/10.1007/s10711-015-0125-1

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