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. |
Contents
| 1 | |
The Theory of Normal Cycles | 7 |
Curves | 13 |
3 | 29 |
4 | 47 |
Elements of Measure Theory | 57 |
Polyhedra | 71 |
Convex Subsets | 77 |
The Steiner Formula for Convex Subsets | 153 |
Tubes Formula | 165 |
Subsets of Positive Reach | 177 |
57 | 182 |
Invariant Forms | 189 |
Curvature Measures of Geometric Sets | 205 |
Second Fundamental Measure | 213 |
Curvature Measures in E² | 221 |
Differential Forms and Densities on EN | 91 |
Background on Riemannian Geometry | 101 |
Riemannian Submanifolds | 109 |
Currents | 121 |
Approximation of the Volume | 129 |
Approximation of the Length of Curves | 139 |
Curvature Measures in E³ | 231 |
Approximation of the Curvature of Curves | 241 |
Approximation of the Curvatures of Surfaces | 249 |
Bibliography | 261 |
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amax approximation Borel subset boundary Chap closely inscribed compact oriented compact subset compute Consequently convergence convex body convex subset curvature measures curvature tensor deduce defined Definition denotes differential forms dimension edges endowed Euler characteristic evaluate following result function Gauss curvature Gauss map Gauss-Bonnet theorem geodesic global Grassmann manifold Hausdorff distance Hausdorff limit Hausdorff topology hypersurface Lebesgue measure Lemma length Lipschitz n-dimensional n-form N-volume normal bundle normal cycle normal vector field null orthogonal projection orthonormal frame outer measure polygonal line polyhedra polyhedron positive reach pr(m principal curvatures proof of Theorem Proposition Quermassintegrale real number resp Riemannian rigid motions second fundamental form Sect sequence Sketch of proof smooth curve smooth submanifold smooth surface Steiner formula subspace tangent space tangent vector tends to infinity transversal integrals triangulation unit normal vector field vertex Voln volume form Voly
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