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. JSTOR 1969800.
- ^ 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.
- ^ 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.
- ^ 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.
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