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
2 | |
The Theory of Normal Cycles | 8 |
Curves | 13 |
Surfaces | 29 |
4 | 47 |
Elements of Measure Theory | 57 |
6 | 71 |
Convex Subsets | 77 |
Currents | 121 |
Approximation of the Volume | 129 |
14 | 139 |
16 | 153 |
23 | 221 |
Curvature Measures in E3 | 231 |
Approximation of the Curvature of Curves | 241 |
Approximation of the Curvatures of Surfaces | 249 |
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angle approximation associated boundary bounded bundle called Chap Chapter classical closely inscribed compact computation Consequently consider constant construction continuous convergence convex body convex subset currents curvature measures curve deal deduce defined Definition denotes differential dimension direct distance edges endowed equals evaluate exists faces formula frame function Gauss curvature geometric give given global Hausdorff implies instance integral introduce invariant length limit manifold mean measure Moreover normal cycle normal vector Note notion obtained oriented orthogonal projection particular plane polygon polyhedra polyhedron principal properties Proposition proved reader real number regular Remark resp respect restriction result Riemannian satisfies Sect sequence shows signed simple smooth smooth curve smooth submanifold smooth surface space subset surface tangent tangent vector tends tensor topology triangulation unit vector field vertex vertices volume
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