Centroids and Equilibrium

A Pure Mathematical Approach
1. Auflage 2027
ISBN: 978-1-041-28057-6
Verlag: Taylor & Francis

Buch, Englisch, 312 Seiten, Format (B × H): 178 mm x 254 mm

Reihe: Discrete Mathematics and Its Applications

Centroids and Equilibrium: A Pure Mathematical Approach investigates the geometry of centroids and static equilibria of convex bodies. Bringing together classical questions and recent developments, it shows how concepts originating in physics and engineering have given rise to a rich variety of problems in pure mathematics.

The book surveys three classical themes in convex geometry related to centroids: Grünbaum’s centroid inequality, the Busemann–Petty centroid inequality, and Ulam’s problem on convex bodies floating in equilibrium in every position. The discussion includes stability results, extensions, and related open questions. Four further chapters focus on static equilibria of convex bodies. They address the classification of bodies by their equilibrium properties, transitions between equilibrium classes and geometric robustness, monostable polyhedra and mechanical complexity, as well as the location and clustering of equilibria and their behavior with respect to a general reference point.

The book is divided into two parts, progressing from an accessible survey to graduate-level proofs. Part I presents the main problems and results, with practice problems at the end of each chapter and prerequisites largely limited to calculus and linear algebra. Part II provides selected proofs using methods from analysis, dynamical systems, and measure theory. The chapters are largely self-contained, allowing readers to explore individual topics independently. With around fifty research problems, the volume offers advanced students and researchers both an introduction to these interconnected areas of convex geometry and a pathway toward current research.

Key Features:

- Gives a unified approach to classical convex geometry inequalities related to centroids and to the theory of equilibria, which are usually treated separately

- Its two-part structure permits the same volume to serve both as an introduction and as an advanced reference

- Ideally suited for both students and researchers of mathematics

Zielgruppe

Academic and Postgraduate

Autoren/Hrsg.

Lángi, Zsolt

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis
• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik

Inhaltsverzeichnis

Part I: Selected topics  1. Preliminaries  2. Grünbaum’s centroid inequality 3. The Busemann-Petty centroid inequality 4. Convex bodies floating in water 5. Classification of convex bodies based on their equilibrium properties 6. Equilibrium points of convex polyhedra 7. The location and clustering of equilibria  8. Equilibria with respect to a general reference point: Problems of concurrent normals  Part II: Selected proofs  9. Selected proofs from Chapter 2 10. Selected proofs from Chapter 3 11. Selected proofs from Chapter 4 12. Selected proofs from Chapter 5 13. Selected proofs from Chapter 6 14. Selected proofs from Chapter 7 15. Selected proofs from Chapter 8 16. Open problems: An overview

Über Autor(innen)

Zsolt Lángi is an associate professor at University of Szeged, a senior research fellow at the University of Dunaújváros, and the Hungarian principal investigator of a Chinese-Hungarian research consortium on lattice geometry and its applications. He received his Ph.D. in mathematics at Eötvös University of Budapest, and also at the University of Calgary. He is particularly interested in geometric extremum problems, and equilibrium points of convex bodies.