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This textbook treats two important and related matters in convex geometry: the quantification of symmetry of a convex set—measures of symmetry—and the degree to which convex sets that nearly minimize such measures of symmetry are themselves nearly symmetric—the phenomenon of stability. By gathering the subject’s core ideas and highlights around Grünbaum’s general notion of measure of symmetry, it paints a coherent picture of the subject, and guides the reader from the basics to the state-of-the-art. The exposition takes various paths to results in order to develop the reader’s grasp of the unity of ideas, while interspersed remarks enrich the material with a behind-the-scenes view of corollaries and logical connections, alternative proofs, and allied results from the literature. Numerous illustrations elucidate definitions and key constructions, and over 70 exercises—with hints and references for the more difficult ones—test and sharpen the reader’s comprehension. The presentation includes: a basic course covering foundational notions in convex geometry, the three pillars of the combinatorial theory (the theorems of Carathéodory, Radon, and Helly), critical sets and Minkowski measure, the Minkowski–Radon inequality, and, to illustrate the general theory, a study of convex bodies of constant width; two proofs of F. John’s ellipsoid theorem; a treatment of the stability of Minkowski measure, the Banach–Mazur metric, and Groemer’s stability estimate for the Brunn–Minkowski inequality; important specializations of Grünbaum’s abstract measure of symmetry, such as Winternitz measure, the Rogers–Shepard volume ratio, and Guo’s Lp -Minkowski measure; a construction by the author of a new sequence of measures of symmetry, the kth mean Minkowski measure; and lastly, an intriguing application to the moduli space of certain distinguished maps from a Riemannian homogeneous space to spheres—illustrating the broad mathematical relevance of the book’s subject.

Geometry --- Mathematics --- Physical Sciences & Mathematics --- Mathematics. --- Convex geometry. --- Discrete geometry. --- Convex and Discrete Geometry. --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Discrete mathematics --- Convex geometry . --- Combinatorial geometry

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The interaction between ergodic theory and discrete groups has a long history and much work was done in this area by Hedlund, Hopf and Myrberg in the 1930s. There has been a great resurgence of interest in the field, due in large measure to the pioneering work of Dennis Sullivan. Tools have been developed and applied with outstanding success to many deep problems. The ergodic theory of discrete groups has become a substantial field of mathematical research in its own right, and it is the aim of this book to provide a rigorous introduction from first principles to some of the major aspects of the theory. The particular focus of the book is on the remarkable measure supported on the limit set of a discrete group that was first developed by S. J. Patterson for Fuchsian groups, and later extended and refined by Sullivan.

Ergodic theory. --- Discrete groups. --- Groups, Discrete --- Discrete mathematics --- Infinite groups --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Discrete groups

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Topological groups. Lie groups --- Number theory --- Automorphic functions --- Discrete groups --- Congresses --- 511 --- -Discrete groups --- -Groups, Discrete --- Infinite groups --- Fuchsian functions --- Functions, Automorphic --- Functions, Fuchsian --- Functions of several complex variables --- -Number theory --- 511 Number theory --- -511 Number theory --- Groups, Discrete --- Congresses. --- Finite groups --- Groupes finis --- Discrete mathematics --- Discrete groups - Congresses --- Automorphic functions - Congresses

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Academic collection --- 512.54 --- Discrete groups --- Geometrical constructions --- Constructions, Geometric --- Constructions, Geometrical --- Geometric constructions --- Geometry --- Groups, Discrete --- Infinite groups --- Groups. Group theory --- Conferences - Meetings --- 512.54 Groups. Group theory --- Discrete mathematics --- Discrete groups - Congresses --- Geometrical constructions - Congresses

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Discrete groups --- Geometry --- 512.54 --- -Geometry --- -#KVIV:BB --- 512.54 Groups. Group theory --- Groups. Group theory --- Mathematics --- Euclid's Elements --- Groups, Discrete --- Infinite groups --- Congresses --- #KVIV:BB --- Discrete mathematics --- Discrete groups - Congresses --- Geometry - Congresses