Classical Topics in Discrete Geometry

By: KÃ¡roly Bezdek

Write A Review

Hardcover | 7 July 2010

At a Glance

Hardcover
180 Pages

Dimensions(cm)
23.39 x 15.6 x 1.12

Hardcover

$99.95

or 4 interest-free payments of $24.99 with

Aims to ship in 7 to 10 business days

Geometry is a classical core part of mathematics which, with its birth, marked the beginning of the mathematical sciences. Thus, not surprisingly, geometry has played a key role in many important developments of mathematics in the past, as well as in present times. While focusing on modern mathematics, one has to emphasize the increasing role of discrete mathematics, or equivalently, the broad movement to establish discrete analogues of major components of mathematics. In this way, the works of a number of outstanding mathema- cians including H. S. M. Coxeter (Canada), C. A. Rogers (United Kingdom), and L. Fejes-T oth (Hungary) led to the new and fast developing eld called discrete geometry. One can brie y describe this branch of geometry as the study of discrete arrangements of geometric objects in Euclidean, as well as in non-Euclidean spaces. This, as a classical core part, also includes the theory of polytopes and tilings in addition to the theory of packing and covering. D- crete geometry is driven by problems often featuring a very clear visual and applied character. The solutions use a variety of methods of modern mat- matics, including convex and combinatorial geometry, coding theory, calculus of variations, di erential geometry, group theory, and topology, as well as geometric analysis and number theory.

Industry Reviews

From the reviews:

"The present volume actually surveys packing and covering problems in Euclidean space and close cousins. ... Bezdek ... surveys the state of the art, best results, and outstanding conjectures for a host of problems. ... Summing Up: Recommended. Academic audiences, upper-division undergraduates through researchers/faculty." (D. V. Feldman, Choice, Vol. 48 (5), January, 2011)

"The book is intended for graduate students interested in discrete geometry. The book provides a road map to the state-of-the-art of several topics in discrete geometry. It can also serve as a textbook for a graduate level course or a seminar. Additionally, the book is extremely current, with many references to as late as 2009-2010 publications." (Alex Bogomolny, The Mathematical Association of America, August, 2010)

"This very interesting monograph contains a selection of topics in discrete geometry, mainly those on which the author and his collaborators have worked. ... The many conjectures and problems to be found throughout the text will serve as an inspiration to many discrete geometers." (Konrad Swanepoel, Zentralblatt MATH, Vol. 1207, 2011)

Shipping

	Standard Shipping	Express Shipping
Metro postcodes:	$9.99	$14.95
Regional postcodes:	$9.99	$14.95
Rural postcodes:	$9.99	$14.95

How to return your order

At Booktopia, we offer hassle-free returns in accordance with our returns policy. If you wish to return an item, please get in touch with Booktopia Customer Care.

Additional postage charges may be applicable.

Defective items

If there is a problem with any of the items received for your order then the Booktopia Customer Care team is ready to assist you.

For more info please visit our Help Centre.

You Can Find This Book In

Non-Fiction Mathematics Geometry

This product is categorised by

Non-Fiction Mathematics Geometry

Preface	p. VII
Classical Topics Revisited
Sphere Packings	p. 3
Kissing Numbers of Spheres	p. 3
One-Sided Kissing Numbers of Spheres	p. 5
On the Contact Numbers of Finite Sphere Packings	p. 6
Lower Bounds for the (Surface) Volume of Voronoi Cells in Sphere Packings	p. 7
On the Density of Sphere Packings in Spherical Containers	p. 12
Upper Bounds on Sphere Packings in High Dimensions	p. 13
Uniform Stability of Sphere Packings	p. 15
Finite Packings by Translates of Convex Bodies	p. 17
Hadwiger Numbers of Convex Bodies	p. 17
One-Sided Hadwiger Numbers of Convex Bodies	p. 18
Touching Numbers of Convex Bodies	p. 19
On the Number of Touching Pairs in Finite Packings	p. 20
Coverings by Homothetic Bodies - Illumination and Related Topics	p. 23
The Illumination Conjecture	p. 23
Equivalent Formulations	p. 24
The Illumination Conjecture in Dimension Three	p. 24
The Illumination Conjecture in High Dimensions	p. 25
On the X-Ray Number of Convex Bodies	p. 28
The Successive Illumination Numbers of Convex Bodies	p. 29
The Illumination and Covering Parameters of Convex Bodies	p. 31
On the Vertex Index of Convex Bodies	p. 32
Coverings by Planks and Cylinders	p. 35
Plank Theorems	p. 35
Covering Convex Bodies by Cylinders	p. 37
Covering Lattice Points by Hyperplanes	p. 39
On Some Strengthenings of the Plank Theorems of Ball and Bang	p. 41
On Partial Coverings by Planks: Bang's Theorem Revisited	p. 43
On the Volume of Finite Arrangements of Spheres	p. 47
The Conjecture of Kneser and Poulsen	p. 47
The Kneser-Poulsen Conjecture for Continuous Contractions	p. 48
The Kneser-Poulsen Conjecture in the Plane	p. 49
Non-Euclidean Kneser-Poulsen-Type Results	p. 51
Alexander's Conjecture	p. 53
Densest Finite Sphere Packings	p. 54
Ball-Polyhedra as Intersections of Congruent Balls	p. 57
Disk-Polygons and Ball-Polyhedra	p. 57
Shortest Billiard Trajectories in Disk-Polygons	p. 57
Blaschke-Lebesgue-Type Theorems for Disk-Polygons	p. 59
On the Steinitz Problem for Ball-Polyhedra	p. 61
On Global Rigidity of Ball-Polyhedra	p. 62
Separation and Support for Spindle Convex Sets	p. 63
Carathéodory- and Steinitz-Type Results	p. 65
Illumination of Ball-Polyhedra	p. 65
The Euler-Poincaré Formula for Ball-Polyhedra	p. 67
Selected Proofs
Selected Proofs on Sphere Packings	p. 71
Proof of Theorem 1.3.5	p. 71
A proof by estimating the surface area of unions of balls	p. 71
On the densest packing of congruent spherical caps of special radius	p. 73
Proof of Theorem 1.4.7	p. 73
The Voronoi star of a Voronoi cell in unit ball packings	p. 73
Estimating the volume of a Voronoi star from below	p. 74
Proof of Theorem 1.4.8	p. 75
Basic metric properties of Voronoi cells in unit ball packings	p. 75
Wedges of types I, II, and III, and truncated wedges of types I, and II	p. 76
The lemma of comparison and a characterization of regular polytopes	p. 79
Volume formulas for (truncated) wedges	p. 80
The integral representation of surface density in (truncated) wedges	p. 81
Truncation of wedges increases the surface density	p. 84
Maximum surface density in truncated wedges of type I	p. 85
An upper bound for the surface density in truncated wedges of type II	p. 86
The overall estimate of surface density in Voronoi cells	p. 88
Proof of Theorem 1.7.3	p. 89
The signed volume of convex polytopes	p. 89
The volume force of convex polytopes	p. 90
Critical volume condition	p. 91
Strictly locally volume expanding convex polytopes	p. 92
From critical volume condition and infinitesimal rigidity to uniform stability of sphere packings	p. 94
Selected Proofs on Finite Packings of Translates of Convex Bodies	p. 95
Proof of Theorem 2.2.1	p. 95
Monotonicity of a special integral function	p. 95
A proof by slicing via the Brunn-Minkowski inequality	p. 96
Proof of Theorem 2.4.3	p. 98
Selected Proofs on Illumination and Related Topics	p. 101
Proof of Corollary 3.4.2 Using Rogers' Classical Theorem on Economical Coverings	p. 101
Proof of Theorem 3.5.2 via the Gauss Map	p. 102
Proof of Theorem 3.5.3 Using Antipodal Spherical Codes of Small Covering Radii	p. 103
Proofs of Theorem 3.8.1 and Theorem 3.8.3	p. 106
From the Banach-Mazur distance to the vertex index	p. 106
Calculating the vertex index of Euclidean balls in dimensions 2 and 3	p. 107
A lower bound for the vertex index using the Blaschke-Santaló inequality and an inequality of Ball and Pajor	p. 112
An upper bound for the vertex index using a theorem of Rudelson	p. 113
Selected Proofs on Coverings by Planks and Cylinders	p. 115
Proof of Theorem 4.1.7	p. 115
On coverings of convex bodies by two planks	p. 115
A proof of the affine plank conjecture of Bang for non-overlapping cuts	p. 116
Proof of Theorem 4.2.2	p. 117
Covering ellipsoids by 1-codimensional cylinders	p. 117
Covering convex bodies by cylinders of given codimension	p. 118
Proof of Theorem 4.5.2	p. 119
Proof of Theorem 4.5.8	p. 119
Selected Proofs on the Kneser-Poulsen Conjecture	p. 121
Proof of Theorem 5.3.2 on the Monotonicity of Weighted Surface Volume	p. 121
Proof of Theorem 5.3.3 on Weighted Surface and Codimension Two Volumes	p. 124
Proof of Theorem 5.3.4 - the Leapfrog Lemma	p. 126
Proof of Theorem 5.4.1	p. 127
The spherical leapfrog lemma	p. 127
Smooth contractions via Schläfli's differential formula	p. 127
Relating higher-dimensional spherical volumes to lower-dimensional ones	p. 128
Putting pieces together	p. 129
Proof of Theorem 5.4.6	p. 130
Monotonicity of the volume of hyperbolic simplices	p. 130
From Andreev's theorem to smooth one-parameter family of hyperbolic polyhedra	p. 133
Selected Proofs on Ball-Polyhedra	p. 135
Proof of Theorem 6.2.1	p. 135
Finite sets that cannot be translated into the interior of a convex body	p. 135
From generalized billiard trajectories to shortest ones	p. 137
Proofs of Theorems 6.6.1, 6.6.3, and 6.6.4	p. 138
Strict separation by spheres of radii at most one	p. 138
Characterizing spindle convex sets	p. 139
Separating spindle convex sets	p. 139
Proof of Theorem 6.7.1	p. 140
On the boundary of spindle convex hulls in terms of supporting spheres	p. 140
From the spherical Carathéodory theorem to an analogue for spindle convex hulls	p. 141
Proof of Theorem 6.8.3	p. 142
On the boundary of spindle convex hulls in terms of normal images	p. 142
On the Euclidean diameter of spindle convex hulls and normal images	p. 143
An upper bound for the illumination number based on a probabilistic approach	p. 144
Schramm's lower bound for the proper measure of polars of sets of given diameter in spherical space	p. 145
An upper bound for the number of sets of given diameter that are needed to cover spherical space	p. 147
The final upper bound for the illumination number	p. 148
Proof of Theorem 6.9.1	p. 148
The CW-decomposition of the boundary of a standard ball-polyhedron	p. 148
On the number of generating balls of a standard ball-polyhedron	p. 149
Basic properties of face lattices of standard ball-polyhedra	p. 150
References	p. 153
Table of Contents provided by Ingram. All Rights Reserved.

Classical Topics in Discrete Geometry

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

More in Geometry

Calculus

3rd Edition

The Maths Book

Big Ideas Simply Explained

The Golden Ratio

Divine Beauty of Mathematics

The Practice of Algebraic Curves

A Second Course in Algebraic Geometry

Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory

Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories

Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory

Volume II: Braided Bimonoidal Categories with Applications

Basic Modern Theory of Linear Complex Analytic $q$-Difference Equations

Mathematical Surveys and Monographs

Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory

Volume III: From Categories to Structured Ring Spectra

Lectures on Differential Geometry

Group Actions and Equivariant Cohomology

Sum Stories Equations and Their Origins

Equations and Their Origins

Introduction to Lie Algebras

Finite and Infinite Dimension

Introduction to Lie Algebras

Finite and Infinite Dimension

The Practice of Algebraic Curves

A Second Course in Algebraic Geometry

Topology at Infinity of Discrete Groups

The Inductive Blockwise Alperin Weight Condition for the Chevalley Group_4(q)$

Geometry, Groups and Mathematical Philosophy

$p$-Adic Hodge Theory for Artin Stacks

Drinfeld Modular Forms of Arbitrary Rank

Lectures on Differential Geometry

Cataland

Why the Fuss?

Twisted Patterson-Sullivan Measures and Applications to Amenability andCoverings

Equivariant Infinite Loop Space Theory

The Space Level Story

Nankai Symposium on Mathematical Dialogues

Celebrating the 110th anniversary of the birth of Prof. S.-S. Chern

This product is categorised by