Entering and Leaving j-Facets

@article{Welzl2001EnteringAL,
  title={Entering and Leaving j-Facets},
  author={Emo Welzl},
  journal={Discrete & Computational Geometry},
  year={2001},
  volume={25},
  pages={351-364}
}
Let S be a set of n points in d-space, no i + 1 points on a common (i − 1)-flat for 1 ≤ i ≤ d. An oriented (d − 1)-simplex spanned by d points in S is called jfacet of S, if there are exactly j points from S on the positive side of its affine hull. We show: (*) For j ≤ n/2 − 2, the total number of (≤ j)-facets (i.e. the number of i-facets with 0 ≤ i ≤ j) in 3-space is maximized in convex position (where these numbers are known). A large part of this presentation is a preparatory review of some… CONTINUE READING
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