On taxicab distance mean functions and their geometric applications: methods, implementations and examples

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.Comment: 26 pages, 6 figures, the paper is based on the plenary lecture presented at Meeting on Tomography and Applications (Discrete Tomography, Neuroscience and Image Reconstruction) 16th Edition, IN MEMORIAM OF CARLA PERI, 2 - 4 May 2022, Mathematics Department, Politecnico di Milano, Milano, Ital

Group of All Taxicab Isometries: A Combinatorial Approach

In this work, we give a more thorough and exhaustive proof of the set of all isometries in taxicab geometry using a combinatorial approach. We show that isometries preserving taxicab distance while leaving the origin fixed are uniquely determined by how they permute the vertices of circles. Then, we use this principle to identify all isometries in taxicab geometry

A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two more references adde