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]
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
^Macbeath, A. M. (September 1952). "A Theorem on Non-Homogeneous Lattices". The Annals of Mathematics. 56 (2): 269–293.
doi:
10.2307/1969800.
JSTOR1969800.
^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.
^
abcBárány, Imre (June 8, 2001). "The techhnique of M-regions and cap-coverings: a survey". Rendiconti di Palermo. 65: 21–38.
^
abcAbdelkader, 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.
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]
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
^Macbeath, A. M. (September 1952). "A Theorem on Non-Homogeneous Lattices". The Annals of Mathematics. 56 (2): 269–293.
doi:
10.2307/1969800.
JSTOR1969800.
^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.
^
abcBárány, Imre (June 8, 2001). "The techhnique of M-regions and cap-coverings: a survey". Rendiconti di Palermo. 65: 21–38.
^
abcAbdelkader, 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.