Affine integral geometry and convex bodies
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 represe...
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1988
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222720_v151_n3_p229_Santalo http://hdl.handle.net/20.500.12110/paper_00222720_v151_n3_p229_Santalo |
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paper:paper_00222720_v151_n3_p229_Santalo2023-06-08T14:48:41Z Affine integral geometry and convex bodies Affine transformations breadth convex bodies invariant densities n‐ellipsoids n‐parallelepipeds n‐simplices parallel hyperplanes r‐flats 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 1988 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222720_v151_n3_p229_Santalo http://hdl.handle.net/20.500.12110/paper_00222720_v151_n3_p229_Santalo |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Affine transformations breadth convex bodies invariant densities n‐ellipsoids n‐parallelepipeds n‐simplices parallel hyperplanes r‐flats |
| spellingShingle |
Affine transformations breadth convex bodies invariant densities n‐ellipsoids n‐parallelepipeds n‐simplices parallel hyperplanes r‐flats Affine integral geometry and convex bodies |
| topic_facet |
Affine transformations breadth convex bodies invariant densities n‐ellipsoids n‐parallelepipeds n‐simplices parallel hyperplanes r‐flats |
| description |
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 |
| title |
Affine integral geometry and convex bodies |
| title_short |
Affine integral geometry and convex bodies |
| title_full |
Affine integral geometry and convex bodies |
| title_fullStr |
Affine integral geometry and convex bodies |
| title_full_unstemmed |
Affine integral geometry and convex bodies |
| title_sort |
affine integral geometry and convex bodies |
| publishDate |
1988 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222720_v151_n3_p229_Santalo http://hdl.handle.net/20.500.12110/paper_00222720_v151_n3_p229_Santalo |
| _version_ |
1768541784739151872 |