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Macbeath region

Brief description on Macbeath Regions From Wikipedia, the free encyclopedia

Macbeath region
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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]

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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
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Definition

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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.

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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
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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
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References

Further reading

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