LEADER 03445nam 2200481 450 001 996490271403316 005 20231110214937.0 010 $a3-031-15127-5 035 $a(CKB)5850000000078333 035 $a(MiAaPQ)EBC7102080 035 $a(Au-PeEL)EBL7102080 035 $a(PPN)264952626 035 $a(EXLCZ)995850000000078333 100 $a20230225d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aConvex cones $egeometry and probability /$fRolf Schneider 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (352 pages) 225 1 $aLecture Notes in Mathematics ;$vv.2319 311 $a3-031-15126-7 320 $aIncludes bibliographical references and index. 327 $aIntro -- Preface -- Contents -- 1 Basic notions and facts -- 1.1 Notation and prerequisites -- 1.2 Incidence algebras -- 1.3 Convex cones -- 1.4 Polyhedra -- 1.5 Recession cones -- 1.6 Valuations -- 1.7 Identities for characteristic functions -- 1.8 Polarity as a valuation -- 1.9 A characterization of polarity -- 2 Angle functions -- 2.1 Invariant measures -- 2.2 Angles -- 2.3 Conic intrinsic volumes and Grassmann angles -- 2.4 Polyhedral Gauss-Bonnet theorems -- 2.5 A tube formula for compact general polyhedra -- 3 Relations to spherical geometry -- 3.1 Basic facts -- 3.2 The gnomonic map -- 3.3 Spherical and conic valuations -- 3.4 Inequalities in spherical space -- 4 Steiner and kinematic formulas -- 4.1 A general Steiner formula for polyhedral cones -- 4.1.1 The local Gaussian Steiner formula -- 4.1.2 The local spherical Steiner formula -- 4.2 Support measures of general convex cones -- 4.3 Kinematic formulas -- 4.4 Concentration of the conic intrinsic volumes -- 4.5 Inequalities and monotonicity properties -- 4.6 Observations about the conic support measures -- 5 Central hyperplane arrangements and induced cones -- 5.1 The Klivans-Swartz formula -- 5.2 Absorption probabilities via central arrangements -- 5.3 Random cones generated by central arrangements -- 5.4 Volume weighted Schl¨afli cones -- 5.5 Typical faces -- 5.6 Intersections of random cones -- 6 Miscellanea on random cones -- 6.1 Random projections -- 6.2 Gaussian images of cones -- 6.3 Wendel probabilities in high dimensions -- 6.4 Donoho-Tanner cones in high dimensions -- 6.5 Cover-Efron cones in high dimensions -- 6.6 Random cones in halfspaces -- 7 Convex hypersurfaces adapted to cones -- 7.1 Coconvex sets -- 7.2 Mixed volumes involving bounded coconvex sets -- 7.3 Wulff shapes in cones -- 7.4 A Minkowski-type existence theorem -- 7.5 A Brunn-Minkowski theorem for coconvex sets. 327 $a7.6 Mixed volumes of general coconvex sets -- 7.7 Minkowski's theorem for general coconvex sets -- 7.8 The cone-volume measure -- 8 Appendix: Open questions -- References -- Notation Index -- Author Index -- Subject Index. 410 0$aLecture Notes in Mathematics 606 $aConvex bodies 606 $aCossos convexos$2thub 608 $aLlibres electrònics$2thub 615 0$aConvex bodies. 615 7$aCossos convexos 676 $a929.605 700 $aSchneider$b Rolf$f1940-$01140742 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996490271403316 996 $aConvex cones$93018498 997 $aUNISA