Geometric Inequalities (Grundlehren der mathematischen Wissenschaften, Vol. 285) (Grundlehren der mathematischen Wissenschaften)

Cover......Page 1 Title......Page 4 Foreword......Page 6 Copyright......Page 5 Table of Contents......Page 10 1.2. Historical Remarks......Page 16 1.3. The Bonnesen Inequality and its Analogues......Page 18 1.4. Examples of Problems with Other Constraints......Page 19 1.5. Affine Isoperimetry......Page 21 2.1. Object of Study and Notations......Page 22 2.2. Isoperimetric Inequalities on Surfaces......Page 26 2.3. Inequalities for Non-simply Connected Domains......Page 29 2.4. Area of Strips Along the Boundary......Page 31 2.5. Radius of the Incircle......Page 34 3. The Main Proofs to 2......Page 35 3.2. Equidistants of the Boundary of a Polyhedron......Page 36 3.3. Proof of the Isoperimetric Inequality......Page 39 3.4. Comparison Lemma......Page 40 3.5. Proofs for 2.4 and 2.3......Page 42 4.1. Sharpenings of Area Estimates of Strips......Page 45 4.2. Proofs to 2.5......Page 47 4.3. Equality Cases......Page 48 4.4. Inequalities when the Combination of a and F is Bounded......Page 53 4.5. Curves in Domains......Page 54 5.1. The Object of Study......Page 56 5.2. Inner Diameter......Page 57 5.3. Shortest Loops and Area Estimates from Below......Page 58 6.1. External Characteristics of Surfaces......Page 63 6.2. Smooth Closed Surfaces in R3......Page 64 6.3. Smooth Surfaces with Boundary......Page 67 6.4. Unboundedness Conditions for Surfaces in E3......Page 68 6.5. Surfaces in R" and CMaps......Page 70 7.1. The Isoperimetric Inequality Involving Total Mean Curvature......Page 73 7.2. Finding the Exact Constants......Page 74 7.3. Isoperimetry on Surfaces with Small Mean Curvature......Page 75 7.4. Estimates of the Area from Below......Page 77 7.5. The Size of Surfaces in Space......Page 79 7.6. Historical Remarks......Page 81 8.1. Inequalities for Compact Sets......Page 83 8.2. Sharper Version and Equality Cases......Page 86 8.3. Inequalities for Arbitrary Sets......Page 89 8.4. Inequalities in Other Spaces which are Additive Groups......Page 90 8.6. Historical Remarks......Page 91 9.2. Symmetrizations......Page 92 9.3. Symmetrization of Neighbourhoods......Page 93 9.4. Multiple Symmetrizations......Page 95 9.6. Arbitrary Sets......Page 97 10.1. Isoperimetric Property of the Euclidean Ball......Page 98 10.2. Isoperimetric Property of the Ball in Spherical and Lobachevsky Spaces......Page 101 10.3. Isoperimetric Inequalities in Finite-Dimensional Normed Spaces......Page 103 10.5. Convex Hulls of Curves in I}B"......Page 105 11.1. Jung's Ball and Other Covering Bodies......Page 106 11.2. Volume Estimates in Terms of Diameter or Width......Page 108 11.3. Volumes of Sets and their Projections......Page 109 11.4. Tetrahedra in Lobachevsky Space......Page 110 12.1. Area of Piecewise Smooth m-Dimensional Surfaces......Page 112 12.2. Why Other Notions of Areas are Needed......Page 113 13.1. Caratheodory Measures......Page 114 13.2. Hausdorff Measures......Page 115 13.3. Eilenberg's Inequality......Page 116 13.4. Coarea Formula......Page 118 14.2. Perimeter and Minkowski Content......Page 121 14.4. The Equality Case in the Classical Isoperimetric Inequality......Page 123 14.5. Perimeter from the Functional Point of View......Page 126 14.7. Perimeter and the Function Space BV......Page 127 15.1. Area of Smooth Maps......Page 129 15.2. Integration over Immersed Manifolds......Page 130 15.3. General Notions of Current and Varifold......Page 131 16.1. Lebesgue Area......Page 135 16.2. Isoperimetric Inequality for Lebesgue Area......Page 136 16.3. Lebesgue Area and Currents......Page 138 16.4. Favard Measures......Page 139 16.5. Integral-Geometrical Areas......Page 140 17.1. Spanning Surfaces for Cycles......Page 141 17.2. Existence of Isoperimetric Spanning Surfaces......Page 143 18.1. The Dido Problem......Page 146 18.3. Isoperimetric Inequalities and Embedding Theorems.......Page 147 18.4. Embedding Theorems for Currents......Page 150 19.1. Mixed Volumes......Page 151 19.2. Properties of Mixed Volumes......Page 152 19.3. Cross-Sectional Measures......Page 153 19.4. Projections......Page 156 20.1. The Main Inequality and Some Consequences......Page 158 20.2. Inequalities of the Isoperimetric Type......Page 159 20.3. About Proofs of the Alexandrov-Fenchel Inequality......Page 160 20.4. Generalizations of the Brunn-Minkowski Theorem......Page 161 20.6. More General Inequalities......Page 162 21.1. Diskant Inequalities......Page 163 21.3. Parallel Sections......Page 166 22.1. Pythagoras Inequality......Page 167 22.2. Bodies with Centre......Page 168 23.1. Volume Estimates for Difference Bodies......Page 169 23.2. Inequalities for Bodies of Rotation......Page 170 24.1. Polar Correspondence......Page 171 24.2. An Example of the Application of the Holder Inequality......Page 172 24.3. Dual Mixed Volumes......Page 173 24.4. Lutwak Inequalities......Page 174 24.5. Inequalities for Polar Bodies......Page 175 24.6. Firey Sums and Related Inclusions and Inequalities......Page 176 24.7. Multilinearity and Inequalities......Page 178 25. Addendum 1. Analogues of Mixed Volumes......Page 179 25.1. The Mixed Volume of Continuous Functions on the Sphere.......Page 180 25.2. Mixed Surface Functions......Page 181 25.3. Curvature Functions......Page 182 25.4. Mixed Discriminants. Permanents......Page 184 25.5. Vector Analogues of Mixed Volumes......Page 185 25.6. Generalizations of the Steiner Decomposition......Page 186 25.7. Centres of Gravity of Curvatures......Page 188 25.8. Mixed Volume as a Distribution......Page 190 25.9. Infinite-Dimensional Case......Page 191 26.1. The Hadwiger Convexity Ring......Page 192 26.2. Arbitrary Sets......Page 193 26.4. Groemer's Linear Space......Page 194 27.1. Outline of the Algebraic Proof of the Alexandrov-Fenchel Inequality......Page 197 27.2. Hyperbolic Quadratic Forms......Page 198 27.3. Remarks on the Theorem Concerning the Number of Roots......Page 200 27.4. Monomials, Monomial Curves, Laurent Polynomials and Their Newton Polyhedra......Page 202 27.5. Intersection of Curves and Hypersurfaces......Page 204 27.6. Riemann Surfaces (Compactification of Algebraic Curves)......Page 206 27.7. Statements of the Theorems and Their Sequence of Proof.......Page 207 27.8. Deduction of the Theorem on the Number of Roots from the Curve Theorem......Page 210 27.9. The Curve Theorem......Page 212 27.10. General (Typical) Systems of Algebraic Equations......Page 216 27.11. Curves on Algebraic Surfaces......Page 218 27.12. Toric Compactification of Spaces......Page 219 27.13. Algebraic Proof of the Alexandrov-Fenchel Inequality.......Page 220 28.1. Mean Curvature......Page 223 28.2. First Variation of Area. Radial Variation and its Applications......Page 224 28.4. Isoperimetric Inequalities Involving Mean Curvature......Page 227 28.5. Embedding Theorems Involving Mean Curvature......Page 231 28.6. The First Variation of the Mass of Currents and Varifolds......Page 232 29.1. The Chern-Lashof Theorem......Page 233 29.3. Symmetric Functions of Principal Curvatures......Page 235 29.4. Immersed Manifolds with Non-Negative Scalar Curvature......Page 237 29.5. Specifications for Low Dimensions......Page 239 30.1. Statement of the Problems......Page 240 30.2. Submanifolds of Small Codimension and Bounded Sectional Curvatures......Page 242 30.3. Hypersurfaces......Page 245 30.4. Maximal Inscribed Ball......Page 246 31.2. The Second Fundamental Form......Page 247 31.3. Jacobi Fields......Page 248 31.4. Second Variation of Length and the Index Form......Page 250 32.1. Minimal Property of Jacobi Fields......Page 251 32.2. Main Lemma......Page 252 32.3. The Rauch Theorem......Page 255 33.1. Jacobians of Exponential Maps......Page 256 33.2. Exponential Maps with Respect to Submanifolds......Page 257 33.3. Jacobians of Exponential Maps with Respect to Submanifolds......Page 259 34.1. Volume of the Neighbourhood of a Submanifold......Page 262 34.2. Linear Isoperimetric Inequalities in Spaces of Negative Curvature......Page 266 34.3. Spherical Isoperimetric Inequality......Page 269 35.1. Volume of Domains with Starlike Boundary......Page 271 35.2. Comparison of Supporting Functions......Page 273 35.3. Volume of Domains and Curvature of the Boundary......Page 279 36.1. The First Variation of Area......Page 280 36.2. Radial Variation......Page 282 36.3. Mean Curvature of Domains with Starlike Boundary......Page 284 36.4. Equidistants and Mean Curvature......Page 288 36.5. The Isoperimetric Inequality Involving Mean Curvature......Page 289 37.1. Volume Estimate from Below......Page 292 37.2. Properties of the Fundamental Group......Page 294 37.3. The Margulis Lemma and the Proof of the Volume Estimate......Page 296 37.4. Proof of Lemma 37.3.2......Page 297 37.5. Estimates to 37.4......Page 303 37.6. Other Inequalities......Page 308 38.1. The Volume of the Topological Cube......Page 309 38.2. Volumes of Cycles and Manifold Volume......Page 312 38.3. Volume and Injectivity Radius......Page 314 Bibliography......Page 315 Author Index......Page 336 Subject Index......Page 341