Catalogue


The cube : a window to convex and discrete geometry /
Chuanming Zong.
imprint
Cambridge, UK ; New York : Cambridge University Press, 2006.
description
x, 174 p. : ill. ; 24 cm.
ISBN
0521855357 (hbk.)
format(s)
Book
Holdings
More Details
imprint
Cambridge, UK ; New York : Cambridge University Press, 2006.
isbn
0521855357 (hbk.)
standard identifier
9780521855358
catalogue key
5896718
 
Includes bibliographical references (p. 166-172) and index.
A Look Inside
Full Text Reviews
Appeared in Choice on 2007-02-01:
Of all the creatures of n-dimensional geometry, the n-cube (the collection of all lists of n numbers between 0 and 1) would seem the simplest one--perhaps the only such structure (aside from the n-simplex, a generalized tetrahedron) that mere mortals might pretend to visualize (when n>3). If analogy with low dimensions provides first intuitions, it also leads one astray. For example, one cannot pave the entire plane with unit squares (i.e., 2-cubes) unless there are two squares somewhere exactly sharing a side; analogous statements (Keller's conjecture) provably hold up to dimension 6, then fail, at least in dimension 8 and higher! Zong (Peking Univ.) in the final chapter explains this circle of recent results; the two prior chapters handle variants. Early chapters survey many properties of n-cubes involving cross-sections, projections, inscribed simplices, triangulations, and convex hulls of vertex sets. The n-cubes unify questions considered, but the techniques underlying the solutions come from analysis, hyperbolic geometry, ring theory, coding theory, and graph theory. Because this book is a love letter to the unity of mathematics, readers can hardly but come to share Zong's undisguised enthusiasm. Perfect for a semester-long topics course! ^BSumming Up: Highly recommended. General readers; upper-division undergraduates through professionals. D. V. Feldman University of New Hampshire
Reviews
Review Quotes
"Because this book is a love letter to the unity of mathematics, readers can hardly but come to share Zong's undisguised enthusiasm...Highly recommended." -- Choice
This item was reviewed in:
Choice, February 2007
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Summaries
Description for Bookstore
This tract has two purposes: to show what is known about the n-dimensional unit cubes and to demonstrate how Analysis, Algebra, Combinatorics, Graph Theory, Hyperbolic Geometry, Number Theory, can be applied to the study of them.
Description for Library
This tract has two purposes: to show what is known about the n-dimensional unit cubes and to demonstrate how Analysis, Algebra, Combinatorics, Graph Theory, Hyperbolic Geometry, Number Theory, can be applied to the study of them. The unit cubes, from any point of view, are among the most important and fascinating objects in an n-dimensional Euclidean space. In this Tract eight topics about the unit cubes are introduced: cross sections, projections, inscribed simplices, triangulations, 0/1 polytopes, Minkowski's conjecture, Furtwangler's conjecture, and Keller's conjecture.
Bowker Data Service Summary
This text, written by Chuanming Zong, demonstrates how analysis, algebra, combinatorics, graph theory, hyperbolic geometry, and number theory, can be applied to the study of unit cubes.
Long Description
Eight topics about the unit cubes are introduced within this textbook: cross sections, projections, inscribed simplices, triangulations, 0/1 polytopes, Minkowski's conjecture, Furtwangler's conjecture, and Keller's conjecture. In particular Chuanming Zong demonstrates how deep analysis like log concave measure and the Brascamp-Lieb inequality can deal with the cross section problem, how Hyperbolic Geometry helps with the triangulation problem, how group rings can deal with Minkowski's conjecture and Furtwangler's conjecture, and how Graph Theory handles Keller's conjecture.
Main Description
This tract has two purposes: to show what is known about the n-dimensional unit cubes and to demonstrate how Analysis, Algebra, Combinatorics, Graph Theory, Hyperbolic Geometry, Number Theory, can be applied to the study of them. The unit cubes, from any point of view, are among the most important and fascinating objects in an n-dimensional Euclidean space. However, our knowledge about them is still quite limited and many basic problems remain unsolved. In this Tract eight topics about the unit cubes are introduced: cross sections, projections, inscribed simplices, triangulations, 0/1 polytopes, Minkowski's conjecture, Furtwangler's conjecture, and Keller's conjecture. In particular the author demonstrates how deep analysis like log concave measure and the Brascamp-Lieb inequality can deal with the cross section problem, how Hyperbolic Geometry helps with the triangulation problem, how group rings can deal with Minkowski's conjecture and Furtwangler's conjecture, and how Graph Theory handles Keller's conjecture.
Table of Contents
Prefacep. vii
Basic notationp. ix
Introductionp. 1
Cross sectionsp. 5
Introductionp. 5
Good's conjecturep. 8
Hensley's conjecturep. 16
Additional remarksp. 28
Projectionsp. 30
Introductionp. 30
Lower bounds and upper boundsp. 32
A symmetric formulap. 39
Combinatorial shapesp. 42
Inscribed simplicesp. 45
Introductionp. 45
Binary matricesp. 48
Upper boundsp. 52
Some particular casesp. 70
Triangulationsp. 73
An examplep. 7
Some special triangulationsp. 75
Smith's lower boundp. 79
Lower-dimensional casesp. 88
0/1 polytopesp. 92
Introductionp. 92
0/1 polytopes and coding theoryp. 93
Classificationp. 99
The number of facetsp. 105
Minkowski's conjecturep. 111
Minkowski's conjecturep. 111
An algebraic versionp. 113
Hajos' proofp. 118
Other versionsp. 125
Furtwangler's conjecturep. 128
Furtwangler's conjecturep. 128
A theorem of Furtwangler and Hajosp. 129
Hajos' counterexamplesp. 131
Robinson's characterizationp. 132
Keller's conjecturep. 150
Keller's conjecturep. 150
A theorem of Keller and Perronp. 151
Corradi and Szabo's criterionp. 155
Lagarias, Mackey, and Shor's counterexamplesp. 163
Referencesp. 166
Indexp. 173
Table of Contents provided by Ingram. All Rights Reserved.

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