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Minkowski Geometry (Encyclopedia of Mathematics and its Applications), by A. C. Thompson
Free Ebook Minkowski Geometry (Encyclopedia of Mathematics and its Applications), by A. C. Thompson
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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.
- Sales Rank: #4146222 in Books
- Brand: Brand: Cambridge University Press
- Published on: 1996-06-28
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x .83" w x 6.14" l, 1.55 pounds
- Binding: Hardcover
- 368 pages
- Used Book in Good Condition
Review
"The author's writing is clear, scholarly and elegant. Each chapter opens with a summary and detailed text follows. An extensive commentary with historical notes closes the chapter. There is a comprehensive bibliography with entries as late as 1995. The printing is accurate and clear as are the many figures, some of which are beautiful." W.J. Firey, Mathematical Reviews
From the Back Cover
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.
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