Strange Phenomena in Convex and Discrete Geometry

Zong, Chuanming; James J. Dudziak (editor)

ISBN 10: 0387947345 / ISBN 13: 9780387947341

Editore: Springer, New York, Berlin, Etc., 1996

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With illustrations. Light shelfwear. ; "Strange Phenomena in Convex and Discrete Geometry" by Chuanming Zong explores intriguing and often counterintuitive properties and problems in the field of geometry. This book examines a variety of unusual phenomena within the context of convex and discrete structures, which are areas known for their complex and sometimes unexpected behavior in mathematical configurations. Zong's work is particularly focused on providing insights into the unique behaviors and characteristics that emerge from the geometric arrangement of figures and points. The book details several key topics, such as the illumination problem, where the challenge is to determine the minimum number of light sources needed to illuminate the entire interior of a given shape; and the Kneser-Poulsen conjecture, which deals with the optimal way to pack spheres so that the distance between their centers is minimized. Throughout the text, Zong uses clear, precise language and provides mathematical proofs to explore these phenomena, making complex topics accessible to those with a solid foundation in mathematics. The book serves both as a scholarly resource for mathematicians and as an intriguing exploration for advanced students engaged in studies of geometric analysis, offering a deep dive into how simple geometric principles can lead to surprising and enlightening conclusions.; Universitext; Octavo; viii, 159 pages.

Codice articolo LCB60987

Dati bibliografici

• Titolo: Strange Phenomena in Convex and Discrete Geometry
• Editore: Springer, New York, Berlin, Etc.
• Data di pubblicazione: 1996
• Lingua: Inglese
• ISBN 10: 0387947345
• ISBN 13: 9780387947341
• Rilegatura: Original decorated wrappers
• Edizione: First Edition; First Impression.
• Condizione: Fine
• Condizione della sovraccoperta: No Dust Jacket

Descrizione articolo

Riassunto

This volume presents some of the most famous problems of convex and discrete geometry as well as their answers. Though covering some of the most recent developments, the book is self-contained and is intended for the reader with little training in mathematics.

Contenuti

1 Borsuk’s Problem.- §1 Introduction.- §2 The Perkal-Eggleston Theorem.- §3 Some Remarks.- §4 Larman’s Problem.- §5 The Kahn-Kalai Phenomenon.- 2 Finite Packing Problems.- §1 Introduction.- §2 Supporting Functions, Area Functions, Minkowski Sums, Mixed Volumes, and Quermassintegrals.- §3 The Optimal Finite Packings Regarding Quermassintegrals.- §4 The L. Fejes Tóth-Betke-Henk-Wills Phenomenon.- §5 Some Historical Remarks.- 3 The Venkov-McMullen Theorem and Stein’s Phenomenon.- §1 Introduction.- §2 Convex Bodies and Their Area Functions.- §3 The Venkov-McMullen Theorem.- §4 Stein’s Phenomenon.- §5 Some Remarks.- 4 Local Packing Phenomena.- §1 Introduction.- §2 A Phenomenon Concerning Blocking Numbers and Kissing Numbers.- §3 A Basic Approximation Result.- §4 Minkowski’s Criteria for Packing Lattices and the Densest Packing Lattices.- §5 A Phenomenon Concerning Kissing Numbers and Packing Densities.- §6 Remarks and Open Problems.- 5 Category Phenomena.- §1 Introduction.- §2 Gruber’s Phenomenon.- §3 The Aleksandrov-Busemann-Feller Theorem.- §4 A Theorem of Zamfirescu.- §5 The Schneider-Zamfirescu Phenomenon.- §6 Some Remarks.- 6 The Busemann-Petty Problem.- §1 Introduction.- §2 Steiner Symmetrization.- §3 A Theorem of Busemann.- §4 The Larman-Rogers Phenomenon.- §5 Schneider’s Phenomenon.- §6 Some Historical Remarks.- 7 Dvoretzky’s Theorem.- §1 Introduction.- §2 Preliminaries.- §3 Technical Introduction.- §4 A Lemma of Dvoretzky and Rogers.- §5 An Estimate for ?V(AV).- §6 ?-nets and ?-spheres.- §7 A Proof of Dvoretzky’s Theorem.- §8 An Upper Bound for M (n, ?).- §9 Some Historical Remarks.- Inedx.