Lectures on Convex Geometry

ISBN-13: 9783030501822 / Angielski / Miękka / 2021 / 287 str.

Daniel Hug; Wolfgang Weil

Recenzje

"'Lectures on convex geometry' is an excellent graduate book about convex geometry. ... The book is very well-organized. ... The presentation is clear, well-composed and illustrated. The problems at the end of each subchapter are carefully selected and revised. The whole text is readable, interesting and easy to learn from. ... the book is excellent and it can serve the studies of the future generation students in convex geometry." (Gergely Kiss, zbMATH 1487.52001, 2022)
"The book is informative, very interesting, and mathematically accessible, and the authors have achieved the purpose that they outline above. As the title suggests, Lectures on convex geometry is well suited to be used as the prescribed textbook for graduate courses in convex geometry; this is because of its pedagogical style and the quality of the exercises. It will also be useful to students intending to pursue a research career in the area ... ." (Daniel John Fresen, Mathematical Reviews, June, 2022)

Spis treści

Preface.- Preliminaries and Notation.- 1. Convex Sets.- 2. Convex Functions.- 3. Brunn-Minkowski Theory.- 4. From Area Measures to Valuations.- 5. Integral Geometric Formulas.-6. Solutions of Selected Exercises.- References.- Index.

O autorze

Prof. Dr. Daniel Hug (1965–) obtained his Ph.D. in Mathematics (1994) and Habilitation (2000) at Univ. Freiburg. He was an assistant Professor at TU Vienna (2000), trained and acted as a High School Teacher (2005–2007), was Professor in Duisburg-Essen (2007), Associate Professor in Karlsruhe (2007–2011), and has been a Professor in Karlsruhe since 2011.

Prof. Dr. Wolfgang Weil (1945–2018) obtained his Ph.D. in Mathematics at Univ. Frankfurt/Main in 1971 and his Habilitation in Freiburg (1976). He was an Assistant Professor in Berlin and Freiburg, Akad. Rat in Freiburg (1978–1980), and was a Professor in Karlsruhe from 1980. He was a Guest Professor in Norman, Oklahoma, USA (1985 and 1990).

Blurb

This book provides a self-contained introduction to convex geometry in Euclidean space.  After covering the basic concepts and results, it develops Brunn–Minkowski  theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including  the isoperimetric inequality.  Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations.  Finally, an introduction to integral-geometric formulas in Euclidean space is provided.  The numerous exercises and the supplementary material at the end of each section form an essential part of the book.

Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry.

Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.