On taxicab distance mean functions and their geometric applications: methods, implementations and examples
It provides a theoretical and computational framework for geometric tomography problems, but is incremental as it extends known concepts to the taxicab metric.
This paper surveys applications of taxicab distance mean functions and generalized conics in geometric tomography, specifically for bisection of focal sets and reconstruction from coordinate X-rays, with implementations in Maple.
A distance mean function measures the average distance of points from the elements of a given set of points (focal set) in the space. The level sets of a distance mean function are called generalized conics. In case of infinite focal points the average distance is typically given by integration over the focal set. The paper contains a survey on the applications of taxicab distance mean functions and generalized conics' theory in geometric tomography: bisection of the focal set and reconstruction problems by coordinate X-rays. The theoretical results are illustrated by implementations in Maple, methods and examples as well.