Titlebook: Generalized Curvatures; Jean-Marie Morvan Book 2008 Springer-Verlag Berlin Heidelberg 2008 Gaussian curvature.Riemannian geometry.Riemanni

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Convex SubsetsThere is an abundant literature on convexity, crucial in many fields of mathematics. We shall mention the basic definitions and some fundamental results (without proof), useful for our topic. In particular, we shall focus on the properties of the volume of a convex body and its boundary. The reader can consult [9, 71, 74, 79] for details.

Scintillations

Differential Forms and Densities on ECurvature measures will be defined by integrating .. Let us introduce their definitions, beginning with exterior algebra in a vector space and continuing with the smooth category. We only give here a brief survey. See [59] for a complete one.

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Approximation of the Length of CurvesWe have seen in Chap. 13 that the length of a curve is classically defined as the supremum of the lengths of polygonal lines inscribed in it. Our purpose here is to compare the length of a given smooth curve with the length of a curve close to it, or more precisely with the length of a polygonal line inscribed in it.

杀菌剂

Tubes FormulaIn Chap. 16, we have seen that the volume of the parallel body of a convex body with smooth boundary is a polynomial whose coefficients depend on the second fundamental form of the boundary. This formula has been generalized by Weyl [82] for the volume of tubes around any smooth submanifold in E., with or without boundary.

wall-stress

Subsets of Positive ReachIn previous chapters, we have seen that it is possible to define . which describe the global shape of two classes of subsets of E., namely the convex bodies and the smooth submanifolds. A good challenge is to find larger classes of subsets on which a more general theory holds. In 1958, Federer [43] made a major advance in two directions:

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Stefan M. Duma Ph.D.,Steven Rowson Ph.D.ate precisely what we mean by a geometric quantity. Consider a subset . of points of the .-dimensional Euclidean space E., endowed with its standard scalar product < ., . >. Let . be the group of rigid motions of E.. We say that a quantity .(.) associated to . is . if the corresponding quantity .[.(

Cytology