Lectures on Discrete Geometry

Lectures on Discrete Geometry presents core topics in geometric combinatorics and discrete geometry, based on graduate-level course materials. The text covers fundamental concepts including convex polytopes, arrangements, and geometric graphs. The book progresses through increasingly complex geometric constructions and theorems, introducing key tools and techniques used in discrete geometry research. Each chapter contains detailed proofs and exercises to reinforce understanding. Topics include the Upper Bound Theorem, ham-sandwich cuts, range searching data structures, epsilon-nets, and geometric discrepancy. The material draws connections between combinatorial geometry and other mathematical disciplines like linear programming and topology. The text serves as both an introduction to discrete geometry research and a reference for established theorems and methods in the field. Its systematic approach illustrates the progression from basic principles to advanced geometric concepts.

Readers value this textbook as a clear introduction to discrete geometry for graduate students and researchers. Reviews note the systematic approach and rigorous proofs. Liked: - Clear explanations of complex concepts - Comprehensive coverage of discrete geometry topics - Helpful exercises with varying difficulty levels - Strong focus on computational aspects Disliked: - Some proofs described as overly terse - Advanced prerequisites needed (topology, linear algebra) - Limited coverage of certain specialized topics - High price point for textbook Ratings: Goodreads: 4.0/5 (5 ratings) Amazon: 5/5 (2 ratings) One mathematics PhD student on Mathematics Stack Exchange praised "the straightforward presentation style and natural progression of topics." A reviewer on Amazon noted it "requires significant mathematical maturity" but serves as "an excellent reference for researchers." Limited review data exists online for this specialized academic text, with most discussion occurring in academic forums and course syllabi references.

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🔷 Jiří Matoušek was a renowned Czech mathematician who made significant contributions to computational geometry, discrete mathematics, and topological methods in combinatorics before his passing in 2015. 🔷 The book builds upon lecture notes from courses taught at Charles University in Prague and ETH Zürich, two of Europe's most prestigious institutions for mathematical research. 🔷 Discrete geometry, the subject of this book, has deep connections to linear programming, optimization algorithms, and computer graphics - making it essential for modern computational applications. 🔷 The text includes the celebrated Kővári–Sós–Turán theorem, which established fundamental limits on the number of edges in bipartite graphs and has applications in incidence geometry. 🔷 Published as part of Springer's Graduate Texts in Mathematics series in 2002, it quickly became a standard reference for both discrete and computational geometry, bridging pure mathematics and computer science applications.