LEADER 04075nam 22006015 450 001 9910300258703321 005 20200629123829.0 010 $a3-319-23733-0 024 7 $a10.1007/978-3-319-23733-6 035 $a(CKB)3710000000521709 035 $a(SSID)ssj0001585301 035 $a(PQKBManifestationID)16264703 035 $a(PQKBTitleCode)TC0001585301 035 $a(PQKBWorkID)14864918 035 $a(PQKB)11696121 035 $a(DE-He213)978-3-319-23733-6 035 $a(MiAaPQ)EBC6312059 035 $a(MiAaPQ)EBC5579358 035 $a(Au-PeEL)EBL5579358 035 $a(OCoLC)932168963 035 $a(PPN)190532122 035 $a(EXLCZ)993710000000521709 100 $a20151126d2015 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aMeasures of Symmetry for Convex Sets and Stability /$fby Gabor Toth 205 $a1st ed. 2015. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2015. 215 $a1 online resource (XII, 278 p. 65 illus., 1 illus. in color.) 225 1 $aUniversitext,$x0172-5939 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-319-23732-2 327 $aFirst Things First on Convex Sets -- Affine Diameters and the Critical Set -- Measures of Stability and Symmetry -- Mean Minkowski Measures. 330 $aThis textbook treats two important and related matters in convex geometry: the quantification of symmetry of a convex set?measures of symmetry?and the degree to which convex sets that nearly minimize such measures of symmetry are themselves nearly symmetric?the phenomenon of stability. By gathering the subject?s core ideas and highlights around Grünbaum?s general notion of measure of symmetry, it paints a coherent picture of the subject, and guides the reader from the basics to the state-of-the-art. The exposition takes various paths to results in order to develop the reader?s grasp of the unity of ideas, while interspersed remarks enrich the material with a behind-the-scenes view of corollaries and logical connections, alternative proofs, and allied results from the literature. Numerous illustrations elucidate definitions and key constructions, and over 70 exercises?with hints and references for the more difficult ones?test and sharpen the reader?s comprehension. The presentation includes: a basic course covering foundational notions in convex geometry, the three pillars of the combinatorial theory (the theorems of Carathéodory, Radon, and Helly), critical sets and Minkowski measure, the Minkowski?Radon inequality, and, to illustrate the general theory, a study of convex bodies of constant width; two proofs of F. John?s ellipsoid theorem; a treatment of the stability of Minkowski measure, the Banach?Mazur metric, and Groemer?s stability estimate for the Brunn?Minkowski inequality; important specializations of Grünbaum?s abstract measure of symmetry, such as Winternitz measure, the Rogers?Shepard volume ratio, and Guo?s Lp -Minkowski measure; a construction by the author of a new sequence of measures of symmetry, the kth mean Minkowski measure; and lastly, an intriguing application to the moduli space of certain distinguished maps from a Riemannian homogeneous space to spheres?illustrating the broad mathematical relevance of the book?s subject. 410 0$aUniversitext,$x0172-5939 606 $aConvex geometry 606 $aDiscrete geometry 606 $aConvex and Discrete Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21014 615 0$aConvex geometry. 615 0$aDiscrete geometry. 615 14$aConvex and Discrete Geometry. 676 $a510 700 $aToth$b Gabor$4aut$4http://id.loc.gov/vocabulary/relators/aut$061811 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910300258703321 996 $aMeasures of symmetry for convex sets and stability$91522799 997 $aUNINA