Horst Martini – författare

This textbook focuses on the geometry of circles, spheres, and spherical geometry. Various classic themes are used as introductory and motivating topics.

The book begins very simply for the reader in the first chapter discussing the notions of inversion and stereographic projection. Here, various classical topics and theorems such as Steiner cycles, inversion, Soddy''s hexlet, stereographic projection and Poncelet''s porism are discussed. The book then delves into Bend formulas and the relation of radii of circles, focusing on Steiner circles, mutually tangent four circles in the plane and other related notions. Next, some fundamental concepts of graph theory are explained. The book then proceeds to explore orthogonal-cycle representation of quadrangulations, giving detailed discussions of the Brightwell-Scheinerman theorem (an extension of the Koebe-Andreev-Thurston theorem), Newton’s 13-balls-problem, Casey’s theorem (an extension of Ptolemy’s theorem) and its generalizations. The remainder of the book is devoted to spherical geometry including a chapter focusing on geometric probability on the sphere.

The book also contains new results of the authors and insightful notes on the existing literature, bringing the reader closer to the research front. Each chapter concludes with related exercises of varying levels of difficulty. Solutions to selected exercises are provided. 

This book is suitable to be used as textbook for a geometry course or alternatively as basis for a seminar for both advanced undergraduate and graduate students alike.

This book provides the solutions to all 347 exercises contained in the text Convexity from the Geometric Point of View, published in the same Cornerstones series. All these exercises are restated and numbered analogously to those in the original text. The corresponding solutions follow each exercise.   Besides the discussion of all solutions, some additional facts about the main text are sprinkled throughout. Sections of further reading are posted to the ends of each chapter supplying the reader with background literature to selected notions and tools that play a role in the exercises and/or solutions to the chapter.

The original text gives a comprehensive introduction to the “common core” of convex geometry and is suitable as a primary text for courses in convex geometry and in discrete geometry (including polytopes). Additionally, it can be used as a single reference for a complete introduction to convex geometry. The content coverage is sufficiently broad that the reader may gain a glimpse of the entire breadth of the field, various subfields, and interesting connections to neighboring disciplines. Mainly directed to graduate and advanced undergraduates, the original text is self-contained in such a way that it can be read by anyone who has standard undergraduate knowledge of analysis and of linear algebra. The same is true for this book of solutions.