Tangent Segments And Secant Segments In Minkowski Planes

Minkowski Geometry plays an important role in Banach Space Theory and Local Theory of Banach Spaces, and it has many applications in Optimization, Operation Reasearch, Combinatorial Geometry, Discrete and Computational Geometry. Therefore there is a good reason to study it carefully.First we survey main results in this research field, show the objective of this research project and the main problems that we will study.Then preliminaries of Minkowski Geometry, including orthogonality types, Radon planes, symmetric Minkowski planes, strict convexity, smoothness, and triangle inequality, which are important for our study, are collected.Tangent segment and related characterization of inner product spaces are discussed, an existing result is strengthened and a condition to ensure the Secant segment Theorem holding in Minkowski spaces is obtained. Furthermore, the relation between tangent segment and strict convexity and smoothness, and the relation between secant segment and triangle inequality are studied.