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A Geometric Lower Bound Theorem

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Abstract

We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality. Further, for C 2-convex bodies, asymptotically tight lower bounds on the g-numbers of the approximating polytopes are given, in terms of their Hausdorff distance from the convex body.

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Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Karim Adiprasito & Eran Nevo

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195-4350, USA

    José Alejandro Samper

Authors
  1. Karim Adiprasito
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  2. Eran Nevo
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  3. José Alejandro Samper
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Correspondence to Karim Adiprasito.

Additional information

Dedicated to Gil Kalai on the occasion of his 60th birthday.

K. A. Adiprasito acknowledges support by a Minerva postdoctoral fellowship of the Max Planck Society, and NSF Grant DMS 1128155. Research of E. Nevo was partially supported by Israel Science Foundation grants ISF-805/11 and ISF-1695/15. J. A. Samper thanks Isabella Novik for the research assistant positions funded through NSF Grants DMS-1069298 and DMS-1361423.

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Adiprasito, K., Nevo, E. & Samper, J.A. A Geometric Lower Bound Theorem. Geom. Funct. Anal. 26, 359–378 (2016). https://doi.org/10.1007/s00039-016-0363-x

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  • DOI: https://doi.org/10.1007/s00039-016-0363-x

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