@article{746188,
      author = {Toth, Gabor.},
      url = {http://library.usi.edu/record/746188},
      title = {Measures of symmetry for convex sets and stability  [electronic resource] /},
      abstract = {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.},
      doi = {https://doi.org/10.1007/978-3-319-23733-6},
      recid = {746188},
      pages = {1 online resource (xii, 278 pages) :},
}