Centroids and Equilibrium

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 separatelyIts two-part structure permits the same volume to serve both as an introduction and as an advanced referenceIdeally suited for both students and researchers of mathematics