The Macbeath region around a point x in a convex body K and the scaled Macbeath region around a point x in a convex body K

In mathematics, a Macbeath region is an explicitly defined region in convex analysis on a bounded convex subset of d-dimensional Euclidean space . The idea was introduced by Alexander Macbeath (1952)[1] and dubbed by G. Ewald, D. G. Larman and C. A. Rogers in 1970.[2] Macbeath regions have been used to solve certain complex problems in the study of the boundaries of convex bodies.[3] Recently they have been used in the study of convex approximations and other aspects of computational geometry.[4][5]

Definition

Let K be a bounded convex set in a Euclidean space. Given a point x and a scaler λ the λ-scaled the Macbeath region around a point x is:

The scaled Macbeath region at x is defined as:

This can be seen to be the intersection of K with the reflection of K around x scaled by λ.

Example uses

  • Macbeath regions can be used to create approximations, with respect to the Hausdorff distance, of convex shapes within a factor of combinatorial complexity of the lower bound.[5]
  • Macbeath regions can be used to approximate balls in the Hilbert metric, e.g. given any convex K, containing an x and a then:[4][6]
  • Dikin’s Method

Properties

  • The is centrally symmetric around x.
  • Macbeath regions are convex sets.
  • If and then .[3][4] Essentially if two Macbeath regions intersect, you can scale one of them up to contain the other.
  • If some convex K in containing both a ball of radius r and a half-space H, with the half-space disjoint from the ball, and the cap of our convex set has a width less than or equal to , we get for x, the center of gravity of K in the bounding hyper-plane of H.[3]
  • Given a convex body in canonical form, then any cap of K with width at most then , where x is the centroid of the base of the cap.[5]
  • Given a convex K and some constant , then for any point x in a cap C of K we know . In particular when , we get .[5]
  • Given a convex body K, and a cap C of K, if x is in K and we get .[5]
  • Given a small and a convex in canonical form, there exists some collection of centrally symmetric disjoint convex bodies and caps such that for some constant and depending on d we have:[5]
    • Each has width , and
    • If C is any cap of width there must exist an i so that and

References

  1. Macbeath, A. M. (September 1952). "A Theorem on Non-Homogeneous Lattices". The Annals of Mathematics. 56 (2): 269–293. doi:10.2307/1969800. JSTOR 1969800.
  2. Ewald, G.; Larman, D. G.; Rogers, C. A. (June 1970). "The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space". Mathematika. 17 (1): 1–20. doi:10.1112/S0025579300002655.
  3. 1 2 3 Bárány, Imre (June 8, 2001). "The techhnique of M-regions and cap-coverings: a survey". Rendiconti di Palermo. 65: 21–38.
  4. 1 2 3 Abdelkader, Ahmed; Mount, David M. (2018). "Economical Delone Sets for Approximating Convex Bodies". 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). 101: 4:1–4:12. doi:10.4230/LIPIcs.SWAT.2018.4.
  5. 1 2 3 4 5 6 Arya, Sunil; da Fonseca, Guilherme D.; Mount, David M. (December 2017). "On the Combinatorial Complexity of Approximating Polytopes". Discrete & Computational Geometry. 58 (4): 849–870. arXiv:1604.01175. doi:10.1007/s00454-016-9856-5. S2CID 1841737.
  6. Vernicos, Constantin; Walsh, Cormac (2021). "Flag-approximability of convex bodies and volume growth of Hilbert geometries". Annales Scientifiques de l'École Normale Supérieure. 54 (5): 1297–1314. arXiv:1809.09471. doi:10.24033/asens.2482. S2CID 53689683.

Further reading

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