The VC-Dimension of K-Vertex D-Polytopes


In this short note, we show that the VC-dimension of the class of k-vertex polytopes in Rd is at most 8d2k log2k, answering an old question of Long and Warmuth. We also show that it is at least 1/3kd.

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We thank Lee-Ad Gottlieb, Aryeh Kontorovich and Gabriel Nivasch for sharing the question and for several enlightening discussions. We also thank the referee for his useful comments, which, in particular, significantly improved the presentation of the lower bound.

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Correspondence to Andrey Kupavskii.

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The author acknowledges the financial support from the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant no 075-15-2019-1926.

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Kupavskii, A. The VC-Dimension of K-Vertex D-Polytopes. Combinatorica 40, 869–874 (2020).

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Mathematics Subject Classification (2010)

  • 52B05
  • 52B55