## Generalized CurvaturesThe central object of this book is the measure of geometric quantities describing N a subset of the Euclidean space (E ,), endowed with its standard scalar product. Let us state precisely what we mean by a geometric quantity. Consider a subset N S of points of the N-dimensional Euclidean space E , endowed with its standard N scalar product. LetG be the group of rigid motions of E . We say that a 0 quantity Q(S) associated toS is geometric with respect toG if the corresponding 0 quantity Q[g(S)] associated to g(S) equals Q(S), for all g?G . For instance, the 0 diameter ofS and the area of the convex hull ofS are quantities geometric with respect toG . But the distance from the origin O to the closest point ofS is not, 0 since it is not invariant under translations ofS. It is important to point out that the property of being geometric depends on the chosen group. For instance, ifG is the 1 N group of projective transformations of E , then the property ofS being a circle is geometric forG but not forG , while the property of being a conic or a straight 0 1 line is geometric for bothG andG . This point of view may be generalized to any 0 1 subsetS of any vector space E endowed with a groupG acting on it. |

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### Contents

2 | |

3 | |

4 | |

5 | |

6 | |

On Volume | 7 |

The Theory of Normal Cycles | 8 |

Applications to Curves and Surfaces | 10 |

123 Three Theorems | 124 |

On Volume | 128 |

Approximation of the Volume | 129 |

132 A General Evaluation Theorem for the Volume | 131 |

133 An Approximation Result | 133 |

134 A Convergence Theorem for the Volume | 135 |

1342 Statement of the Theorem | 137 |

Approximation of the Length of Curves | 139 |

Motivations | 12 |

Motivation Curves | 13 |

212 The General Deﬁnition | 14 |

213 The Length of a C¹Curve | 15 |

214 An Obvious Convergence Result | 16 |

22 The Curvature of a Curve | 17 |

222 The Global or Total Curvature | 19 |

23 The Gauss Map of a Curve | 21 |

24 Curves in E² | 22 |

243 The Signed Curvature of a Smooth Plane Curve | 24 |

244 The Signed Curvature of a Plane Polygon | 26 |

245 Signed Curvature and Topology | 27 |

25 Conclusion | 28 |

Motivation Surfaces | 29 |

313 Warning The Lantern of Schwarz | 30 |

32 The Pointwise Gauss Curvature | 33 |

322 Gauss Curvature and Geodesic Triangles | 34 |

323 The Angular Defect of a Vertex of a Polyhedron | 36 |

324 Warning A Negative Result | 37 |

325 Warning The Pointwise Gauss Curvature of a Closed Surface | 39 |

326 Warning A Negative Result Concerning the Approximation by Quadrics | 40 |

33 The Gauss Map of a Surface | 41 |

332 The Gauss Map of a Polyhedron | 42 |

34 The Global Gauss Curvature | 43 |

35 The Volume | 44 |

Background Metrics and Measures | 46 |

Distance and Projection | 47 |

42 The Projection Map | 49 |

43 The Reach of a Subset | 52 |

44 The Voronoi Diagrams | 55 |

Elements of Measure Theory | 57 |

512 Measures | 58 |

514 Signed Measures | 59 |

515 Borel Measures | 60 |

522 Integral of Measurable Functions | 61 |

53 The Standard Lebesgue Measure on Eᴺ | 62 |

531 Lebesgue Outer Measure on R and Eᴺ | 63 |

532 Lebesgue Measure on R and Eᴺ | 64 |

54 Hausdorff Measures | 65 |

55 Area and Coarea Formula | 66 |

56 Radon Measures | 67 |

Background Polyhedra and Convex Subsets | 69 |

Polyhedra | 71 |

62 Euler Characteristic | 74 |

63 Gauss Curvature of a Polyhedron | 75 |

Convex Subsets | 77 |

712 The Support Function | 79 |

713 The Volume of Convex Bodies | 80 |

72 Differential Properties of the Boundary | 81 |

73 The Volume of the Boundary of a Convex Body | 82 |

74 The Transversal Integral and the Hadwiger Theorem | 84 |

742 Transversal Integral | 85 |

743 The Hadwiger Theorem | 86 |

Background Classical Tools in Differential Geometry | 90 |

Differential Forms and Densities on Eᴺ | 91 |

812 Integration of NDifferential Forms on Eᴺ | 93 |

82 Densities | 94 |

822 Integration of Densities on Eᴺ and the Associated Measure | 95 |

Measures on Manifolds | 97 |

92 Density and Measure on a Manifold | 98 |

93 The Fubini Theorem on a Fiber Bundle | 99 |

Background on Riemannian Geometry | 101 |

102 Properties of the Curvature Tensor | 102 |

103 Connexion Forms and Curvature Forms | 103 |

105 The GaussBonnet Theorem | 104 |

107 The Grassmann Manifolds | 105 |

1072 The Grassmann Manifold GNk | 106 |

1073 The Grassmann Manifolds AGNk and AGᴼNk | 107 |

Riemannian Submanifolds | 109 |

112 The Volume of a Submanifold | 112 |

113 Hypersurfaces in Eᴺ | 113 |

1132 kthMean Curvature of a Hypersurface | 114 |

114 Submanifolds in Eᴺ of Any Codimension | 115 |

1142 kthMean Curvatures in Large Codimension | 116 |

1144 The GaussCodazziRicci Equations | 117 |

115 The Gauss Map of a Submanifold | 118 |

Currents | 121 |

122 Rectiﬁable Currents | 122 |

142 An Approximation by a Polygonal Line | 140 |

Approximation of the Area of Surfaces | 143 |

152 Triangulations | 144 |

1522 Geometric Invariant Associated to a Triangulation | 145 |

154 A Bound on the Deviation Angle | 146 |

1542 Proof of Theorem 45 | 147 |

155 Approximation of the Area of a Smooth Surface by the Area of a Triangulation | 150 |

The Steiner Formula | 151 |

The Steiner Formula for Convex Subsets | 153 |

Segments Discs and Balls | 155 |

163 Convex Bodies in Eᴺ Whose Boundary is a Polyhedron | 158 |

164 Convex Bodies with Smooth Boundary | 159 |

165 Evaluation of the Quermassintegrale by Means of Transversal Integrals | 161 |

166 Continuity of the 𝜱k | 162 |

167 An Additivity Formula | 164 |

Tubes Formula | 165 |

172 The Tubes Formula of Weyl 1939 | 168 |

1722 Intrinsic Character of the Mk | 170 |

173 The Euler Characteristic | 171 |

175 Transversal Integrals | 172 |

176 On the Differentiability of the Immersions | 174 |

Subsets of Positive Reach | 177 |

182 The Steiner Formula | 180 |

183 Curvature Measures | 182 |

185 The Problem of Continuity of the 𝜱k | 184 |

186 The Transversal Integrals | 186 |

The Theory of Normal Cycles | 187 |

Invariant Forms | 189 |

192 Invariant Differential Forms on Eᴺ x Sᴺ¹ | 190 |

193 Examples in Low Dimensions | 192 |

The Normal Cycle | 193 |

2011 Normal Cycle of a Smooth Submanifold | 194 |

2013 Normal Cycle of a Polyhedron | 195 |

2014 Normal Cycle of a Subanalytic Set | 196 |

203 A Convergence Theorem | 198 |

2031 Boundness of the Mass of Normal Cycles | 199 |

204 Approximation of Normal Cycles | 200 |

Curvature Measures of Geometric Sets | 205 |

2111 The Case of Smooth Submanifolds | 206 |

2112 The Case of Polyhedra | 208 |

212 Continuity of the Mk | 209 |

213 Curvature Measures of Geometric Sets | 210 |

Second Fundamental Measure | 213 |

222 Second Fundamental Measure Associated to a Geometric Set | 214 |

223 The Case of a Smooth Hypersurface | 215 |

224 The Case of a Polyhedron | 216 |

226 An Example of Application | 217 |

Applications to Curves and Surfaces | 220 |

Curvature Measures in E² | 221 |

2322 The Mass of the Normal Cycle of a Domain in E² | 223 |

233 Plane Curves | 224 |

2332 The Mass of the Normal Cycle of a Curve in E² | 225 |

234 The Length of Plane Curves | 226 |

2342 Polygon Lines | 227 |

2352 Polygon Lines | 228 |

Curvature Measures in E³ | 231 |

2422 The Length of Space Curves | 232 |

2423 The Curvature of Space Curves | 233 |

243 Surfaces and Bounded Domains in E³ | 234 |

2432 The Mass of the Normal Cycle of a Domain in E³ | 235 |

2433 The Curvature Measures of a Domain | 236 |

244 Second Fundamental Measure for Surfaces | 238 |

Approximation of the Curvature of Curves | 241 |

252 Curves in E³ | 242 |

Approximation of the Curvatures of Surfaces | 249 |

262 Approximation by a Triangulation | 250 |

2622 Approximation of the Curvatures | 251 |

2623 Triangulations Closely Inscribed in a Surface | 252 |

On Restricted Delaunay Triangulations | 253 |

2712 The Empty Ball Property | 254 |

2713 Delaunay Triangulation Restricted to a Subset | 255 |

272 Approximation Using a Delaunay Triangulation | 256 |

2723 Convergence of the Normals | 257 |

2724 Convergence of Length and Area | 258 |

261 | |

265 | |

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1-forms Borel subset boundary bounded Chap Chapter closely inscribed compact subset computation Consequently consider convex body convex subset curvature measures curvature tensor dA(m deduce defined Definition Delaunay triangulation denotes differential forms dimension domain edges endowed Euler characteristic evaluate following result Gauss curvature Gauss map Gauss–Bonnet theorem geodesic geometric invariants geometric subset global Grassmann manifold Hausdorff distance Hausdorff limit Hausdorff topology hypersurface Lebesgue measure Lemma length Let Mn n-dimensional N-volume normal bundle normal cycle normal vector field notation null orthogonal projection orthonormal frame outer measure plane curve pointwise polygonal line polyhedra polyhedron positive reach pr(m principal curvatures proof of Theorem Proposition real number resp Riemannian rigid motions second fundamental form second fundamental measure Sect sequence signed curvature smooth curve smooth submanifold smooth surface Steiner Formula tangent vector tends to infinity Transversal Integrals vector field vertex vertices Voronoi

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