Artículo
Abstract:
Given a convex body Q in Euclidean n‐dimensional space, the affine invariant measure of the set of pairs of parallel hyperplanes containing Q is an affine invariant J(Q) of Q, which, for ellipsoids, parallelepipeds and possibly for simplices of any dimensions, is proportional to V−1, where V represents the volume of Q. Consequently, for the kind of bodies mentioned, it is possible to estimate V−1 from its breadth measured in uniform random directions. If the boundary of Q is of class C2, we obtain a set of affine invariants Jh(Q) (h = any integer) which is easily calculable for ellipsoids. In particular, J‐n(Q) coincides with J(Q) and J‐(n+1)(Q) is the affine invariant measure of all pairs of parallel hyperplanes that ‘support’ Q as described by Firey(1972, 1985), Schneider (1978, 1979) and Weil (1979, 1981). For a general convex body Q the values of Jh(Q) cannot be expressed in terms of simple metric invariants (such as volume, surface area, breadth, width) and this justifies the study in the last section of certain inequalities between them. 1988 Blackwell Science Ltd
Registro:
| Documento: | Artículo |
| Título: | Affine integral geometry and convex bodies |
| Autor: | Santaló, L.A. |
| Filiación: | Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria (Nuñez), Buenos Aires, 1428, Argentina |
| Palabras clave: | Affine transformations; breadth; convex bodies; invariant densities; n‐ellipsoids; n‐parallelepipeds; n‐simplices; parallel hyperplanes; r‐flats |
| Año: | 1988 |
| Volumen: | 151 |
| Número: | 3 |
| Página de inicio: | 229 |
| Página de fin: | 233 |
| DOI: | http://dx.doi.org/10.1111/j.1365-2818.1988.tb04683.x |
| Título revista: | Journal of Microscopy |
| Título revista abreviado: | J. Microsc. |
| ISSN: | 00222720 |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222720_v151_n3_p229_Santalo |
Referencias:
- Blaschke, W., (1923) Vorlesungen über Differentialgeometrie II, , Springer, Berlin
- Bonnesen, T., Fenchel, W., (1934) Theorie der konvexen Körper., , Springer, Berlin
- Firey, W., An integral‐geometric meaning of lower order area functions of convex bodies (1972) Mathematika, 19, pp. 205-212
- Firey, W., The integral‐geometric densities of tangent flats to convex bodies. Discrete geometry and convexity (1985) Annals of the New York Academy of Sciences, 440, pp. 92-96
- Petty, C.M., Affine isoperimetric problems. Discrete geometry and convexity (1985) Annals of the New York Academy of Sciences, 440, pp. 113-127
- Santaló, L.A., (1976) Integral Geometry and Geometric Probability., , Addison‐Wesley, Reading, Mass
- Schneider, R., Curvature measures of convex bodies (1978) Ann. Mat. Pura Appl., 116, pp. 101-134
- Schneider, R., (1979) Integralgeometrie., , Vorlesungen, Freiburg
- Süss, W., Eifläche konstanter Affinbreite (1927) Math. Ann., 96, pp. 251-260
- Weil, W., Kinematic integral formulas for convex bodies (1979) Contributions to Geometry, pp. 60-76. , ed. by, J. Tölke, J. M. Wills, Birkhäuser, Basel
- Weil, W., Berührung konvexer Körper durch q‐dimensionale Ebenen (1981) Resultate Math., 4, pp. 84-101
Citas:
---------- APA ----------
(1988) . Affine integral geometry and convex bodies. Journal of Microscopy, 151(3), 229-233.http://dx.doi.org/10.1111/j.1365-2818.1988.tb04683.x
---------- CHICAGO ----------
Santaló, L.A. "Affine integral geometry and convex bodies" . Journal of Microscopy 151, no. 3 (1988) : 229-233.http://dx.doi.org/10.1111/j.1365-2818.1988.tb04683.x
---------- MLA ----------
Santaló, L.A. "Affine integral geometry and convex bodies" . Journal of Microscopy, vol. 151, no. 3, 1988, pp. 229-233.http://dx.doi.org/10.1111/j.1365-2818.1988.tb04683.x
---------- VANCOUVER ----------
Santaló, L.A. Affine integral geometry and convex bodies. J. Microsc. 1988;151(3):229-233.http://dx.doi.org/10.1111/j.1365-2818.1988.tb04683.x
