Projection body (original) (raw)
Тело сечений — конструкция, дающая тело для данного тела евклидова пространства. Определение было дано Лютваком в 1988 году.Эта конструкция сыграла заметную роль в решении задачи Буземана — Петти.
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| dbo:abstract | In convex geometry, the projection body of a convex body in n-dimensional Euclidean space is the convex body such that for any vector , the support function of in the direction u is the (n – 1)-dimensional volume of the projection of K onto the hyperplane orthogonal to u. Minkowski showed that the projection body of a convex body is convex. and used projection bodies in their solution to Shephard's problem. For a convex body, let denote the polar body of its projection body. There are two remarkable affine isoperimetric inequality for this body. proved that for all convex bodies , where denotes the n-dimensional unit ball and is n-dimensional volume, and there is equality precisely for ellipsoids. proved that for all convex bodies , where denotes any -dimensional simplex, and there is equality precisely for such simplices. The intersection body IK of K is defined similarly, as the star body such that for any vector u the radial function of IK from the origin in direction u is the (n – 1)-dimensional volume of the intersection of K with the hyperplane u⊥.Equivalently, the radial function of the intersection body IK is the Funk transform of the radial function of K.Intersection bodies were introduced by . showed that a centrally symmetric star-shaped body is an intersection body if and only if the function 1/| | x |
| dbo:wikiPageExternalLink | https://books.google.com/books%3Fid=UIjyTgEACAAJ | |
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| dbo:wikiPageWikiLink | dbr:American_Journal_of_Mathematics dbr:Geometriae_Dedicata dbr:Convex_body dbr:Convex_geometry dbr:Lp_norm dbr:Funk_transform dbr:Polar_set dbr:Euclidean_space dbr:Hyperplane dbr:Shephard's_problem dbc:Convex_geometry dbr:Advances_in_Mathematics dbr:Busemann–Petty_problem dbr:Support_function dbr:Springer-Verlag | |
| dbp:b | n (en) | |
| dbp:p | p (en) | |
| dbp:wikiPageUsesTemplate | dbt:Citation dbt:Cite_journal dbt:Harvtxt dbt:Su | |
| dct:subject | dbc:Convex_geometry | |
| gold:hypernym | dbr:Body | |
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| rdfs:comment | Тело сечений — конструкция, дающая тело для данного тела евклидова пространства. Определение было дано Лютваком в 1988 году.Эта конструкция сыграла заметную роль в решении задачи Буземана — Петти. (ru) In convex geometry, the projection body of a convex body in n-dimensional Euclidean space is the convex body such that for any vector , the support function of in the direction u is the (n – 1)-dimensional volume of the projection of K onto the hyperplane orthogonal to u. Minkowski showed that the projection body of a convex body is convex. and used projection bodies in their solution to Shephard's problem. For a convex body, let denote the polar body of its projection body. There are two remarkable affine isoperimetric inequality for this body. proved that for all convex bodies , (en) | |
| rdfs:label | Projection body (en) Тело сечений (ru) | |
| owl:sameAs | freebase:Projection body wikidata:Projection body dbpedia-ru:Projection body https://global.dbpedia.org/id/4tpPc | |
| prov:wasDerivedFrom | wikipedia-en:Projection_body?oldid=1105058555&ns=0 | |
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| is dbo:wikiPageRedirects of | dbr:Intersection_bodies dbr:Intersection_body | |
| is dbo:wikiPageWikiLink of | dbr:Shephard's_problem dbr:Intersection_bodies dbr:Intersection_body | |
| is foaf:primaryTopic of | wikipedia-en:Projection_body |