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]
^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.
^Bárány, Imre (June 8, 2001). "The techhnique of M-regions and cap-coverings: a survey". Rendiconti di Palermo. 65: 21–38.