Macbeath region

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.

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.