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First coherent and complete account of this subject in book form
The 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 Motivations 1 Motivation -Curves
1.1 The length of a curve
1.2 The curvature of a curve
1.3 The Gauss map of a curve
1.4 Curves in E2 2 Motivation -Surfaces
2.1 The area of a surface
2.2 The pointwise Gauss curvature
2.3 The Gauss map of a surface
2.4 The global Gauss curvature
2.5 ... and the volume... 3 Distance and Projection
3.1 The distance function
3.2 The projection map
3.3 The reach of a subset
3.4 The Voronoi diagrams
3.5 The medial axis of a subset 4 Elements of Measure Theory
4.1 Outer measures and measures
4.2 Measurable functions and their integrals
4.3 The standard Lebesgue measure on EN
4.4 Hausdorff measures
4.5 Area and co-area formula
4.6 Radon measures
4.7 Convergence of measures 5 Polyhedra
5.1 Definitions and properties of polyhedra
5.2 Euler characteristic
5.3 Gauss curvature of a polyhedron 6 Convex Subsets
6.1 Convex subsets
6.2 Differential properties of the boundary
6.3 The volume of the boundary of a convex body
6.4 The transversal integral and the Hadwiger theorem 7 Differential Forms and Densities on EN
7.1 Differential forms and their integrals
7.2 Densities 8 Measures on Manifolds
8.1 Integration of differential forms
8.2 Density and measure on a manifold
8.3 The Fubini theorem on a fiber bundle 9 Background on Riemannian Geometry
9.1 Riemannian metric and Levi-Civita connexion
9.2 Properties of the curvature tensor
9.3 Connexion forms and curvature forms
9.4 The volume form
9.5 The Gauss-Bonnet theorem
9.6 Spheres and balls
9.7 The Grassmann manifolds 10 Riemannian Submanifolds
10.1 Some generalities on (smooth) submanifolds
10.2Thevolumeofasubmanifold
10.3 Hypersurfaces in EN
10.4 Submanifolds in EN of any codimension
10.5TheGaussmapofasubmanifold..... 140 11 Currents
11.1 Basic definitions and properties on currents
11.2 Rectifiable currents
11.3Three theorems 12 Approximation of the Volume
12.1 Thegeneralframework
12.2 A general evaluation theorem for the volume
12.3 An approximation result
12.4 Aconvergence theorem for the volume 13 Approximation of the Length of Curves
13.1 A general approximation result
13.2 An approximation by a polygonal line 14 Approximation of the Area of Surfaces
14.1 A general approximation of the area
14.2 Triangulations
14.3 Relative height of a triangulation inscribed in a surface
14.4 A bound on the deviation angle
14.5 Approximation of the area of a smooth surface by the
area of a triangulation 15 The Steiner Formula for Convex Subsets
15.1 The Steiner formula for convex bodies (1840)
15.2 Examples:segments,discsandballs
15.3 Convex bodies in EN whose boundary is a polyhedron
15.4 Convex bodies with smooth boundary
15.5 Evaluation of the Quermassintegrale by means of transversal integrals
15.6 Continuity of the k
15.7 Anadditivity formula 16 Tubes Formula
16.1 The Lipschitz-Killingcurvatures
16.2 The tubes formulaofH.Weyl(1939)
16.3 The Eule rcharacteristic
16.4 Partial continuity of the k
16.5 Transversal integrals
16.6 On the differentiability of the immersions 17 Subsets of Positive Reach
17.1 Subsets of positive reach (H. Federer, 1958)
17.2 The Steiner formula
17.3 Curvature measures
17.4 The Euler characteristic
17.5 The problem of continuity of the k
17.6 Thetransversalintegralses 18 Invariant Forms
18.1 Invariant forms on EN × EN
18.2 Invariant differential forms on EN × SN-1
18.3 Examplesinlow dimensions 19 The Normal Cycle
19.1 The notion of a normal cycle
19.2 Existence and uniqueness of the normal cycle
19.3 A convergence theorem
19.4 Approximation of normal