Minkowski Geometry (Encyclopedia of Mathematics and its Applications, Series Number 63)

Minkowski geometry is a non-Euclidean geometry in a finite number of dimensions that is different from elliptic and hyperbolic geometry (and from the Minkowskian geometry of spacetime). Here the linear structure is the same as the Euclidean one but distance is not "uniform" in all directions. Instead of the usual sphere in Euclidean space, the unit ball is a general symmetric convex set. Therefore, although the parallel axiom is valid, Pythagoras' theorem is not. This book begins by presenting the topological properties of Minkowski spaces, including the existence and essential uniqueness of Haar measure, followed by the fundamental metric properties - the group of isometries, the existence of certain bases and the existence of the Lowner ellipsoid. This is followed by characterizations of Euclidean space among normed spaces and a full treatment of two-dimensional spaces. The three central chapters present the theory of area and volume in normed spaces. The author describes the fascinating geometric interplay among the isoperimetrix (the convex body which solves the isoperimetric problem), the unit ball and their duals, and the ways in which various roles of the ball in Euclidean space are divided among them. The next chapter deals with trigonometry in Minkowski spaces and the last one takes a brief look at a number of numerical parameters associated with a normed space, including J. J. Schaffer's ideas on the intrinsic geometry of the unit sphere. Each chapter ends with a section of historical notes and the book ends with a list of 50 unsolved problems. Minkowski Geometry will appeal to students and researchers interested in geometry, convexity theory and functional analysis. Title page......Page 1 Preface......Page 6 Acknowledgements......Page 12 0.1 Linear spaces......Page 13 0.2 Convex sets......Page 18 0.3 Notes......Page 21 1 Norms and norm topologies......Page 24 1.1 Norm topologies......Page 25 1.2 The unique linear topology on ]Rd......Page 38 1.3 The Hahn-Banach theorem......Page 43 1.4 The existence and uniqueness of Haar measure......Page 47 1.5 Notes......Page 53 2 Convex bodies......Page 56 2.1 Separation and support theorems......Page 57 2.2 Support functions and polar reciprocals......Page 59 2.3 Volumes and mixed volumes......Page 64 2.4 Various derived metrics......Page 71 2.5 Approximation of convex sets and the Blaschke selection theorem......Page 75 2.6 Notes......Page 82 3 Comparisons and contrasts with Euclidean space......Page 86 3.1 The Mazur-Ulam theorem......Page 87 3.2 Normality in Minkowski space......Page 88 3.3 The Löwner ellipsoid......Page 91 3.4 Characterizations of Euclidean space......Page 96 3.5 Notes......Page 105 4 Two-dimensional Minkowski spaces......Page 110 4.1 Inscribed regular hexagons and other constructions......Page 111 4.2 Sets of constant width and equichordal sets......Page 117 4.3 Lengths of curves, perimeter of the unit ball......Page 14 4.4 The isoperimetric problem in a Minkowski plane......Page 129 4.5 Isoperimetric inequalities......Page 134 4,6 Transversality......Page 136 4.7 Radon curves......Page 138 4.8 Notes......Page 140 5 The concept of area and content......Page 146 5.1 Requirements and examples......Page 148 5.2 The role of the function σ_B......Page 152 5.3 The properties and the normalization of I......Page 156 5.4 The isoperimetrices that arise from Examples 5.1 4......Page 161 5.5 Further properties of I......Page 182 5.6 Notes......Page 193 6.1 The convexity of the area function σ......Page 197 6.2 Properties of the mapping I......Page 205 6.3 Cauchy's formula for surface areas......Page 211 6.4 Integral geometry in Minkowski spaces......Page 215 6.5 Bound for the surface area of B......Page 222 6.6 Miscellaneous properties......Page 225 6.7 Notes......Page 232 7.1 The convexity of the area function C'f......Page 238 7.2 Properties of the mapping I......Page 242 7.3 Area and Hausdorff measures......Page 246 7.4 Bound for the surface area of B......Page 251 7.5 Notes......Page 254 8.1 The functions cm and sm......Page 260 8.2 The function α......Page 267 8.3 Trigonometric formulas......Page 269 8.4 Differentiation of the trigonometric functions......Page 273 8.5 Notes......Page 280 9 Various numerical parameters......Page 284 9.1 Projection constants......Page 285 9.2 Macphail's constant......Page 292 9.3 The inner metric......Page 295 9.4 The girth, perimeter, inner radius and inner diameter of X......Page 297 9.5 Five examples in R^d......Page 302 9.6 Relationships with the Banach-Mazur distance and extreme values......Page 309 9.7 Notes......Page 313 10 Fifty problems......Page 316 References......Page 321 Notation index......Page 339 Author index......Page 343 Subject index......Page 347 Minkowski geometry is a type of non-Euclidean geometry in a finite number of dimensions in which distance is not 'uniform' in all directions. This book presents the first comprehensive treatment of Minkowski geometry since the 1940s. The author begins by describing the fundamental metric properties and the topological properties of existence of Minkowski space. This is followed by a treatment of two-dimensional spaces and characterisations of Euclidean space among normed spaces. The central three chapters present the theory of area and volume in normed spaces, a fascinating geometrical interplay among the various roles of the ball in Euclidean space. Later chapters deal with trigonometry and differential geometry in Minkowski spaces. The book ends with a brief look at J. J. Schaffer's ideas on the intrinsic geometry of the unit sphere. Minkowski Geometry will appeal to students and researchers interested in geometry, convexity theory and functional analysis This is a comprehensive treatment of Minkowski geometry. The author begins by describing the fundamental metric properties and the topological properties of existence of Minkowski space. This is followed by a treatment of two-dimensional spaces and characterizations of Euclidean space among normed spaces. The central three chapters present the theory of area and volume in normed spaces--a fascinating geometrical interplay among the various roles of the ball in Euclidean space. Later chapters deal with trigonometry and differential geometry in Minkowski spaces. The book ends with a brief look at J. J. Schaffer's ideas on the intrinsic geometry of the unit sphere.