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- Shephard's_problem abstract "In mathematics, Shephard's problem, is the following geometrical question asked by Geoffrey Colin Shephard (1964): if K and L are centrally symmetric convex bodies in n-dimensional Euclidean space such that whenever K and L are projected onto a hyperplane, the volume of the projection of K is smaller than the volume of the projection of L, then does it follow that the volume of K is smaller than that of L?In this case, "centrally symmetric" means that the reflection of K in the origin, −K, is a translate of K, and similarly for L. If πk : Rn → Πk is a projection of Rn onto some k-dimensional hyperplane Πk (not necessarily a coordinate hyperplane) and Vk denotes k-dimensional volume, Shephard's problem is to determine the truth or falsity of the implicationVk(πk(K)) is sometimes known as the brightness of K and the function Vk o πk as a (k-dimensional) brightness function.In dimensions n = 1 and 2, the answer to Shephard's problem is "yes". In 1967, however, Petty and Schneider showed that the answer is "no" for every n ≥ 3. The solution of Shephard's problem requires Minkowski's first inequality for convex bodies.".
- Shephard's_problem wikiPageID "10614436".
- Shephard's_problem wikiPageRevisionID "572421208".
- Shephard's_problem authorlink "Geoffrey Colin Shephard".
- Shephard's_problem first "Geoffrey Colin".
- Shephard's_problem hasPhotoCollection Shephard's_problem.
- Shephard's_problem last "Shephard".
- Shephard's_problem year "1964".
- Shephard's_problem subject Category:Convex_analysis.
- Shephard's_problem subject Category:Convex_geometry.
- Shephard's_problem comment "In mathematics, Shephard's problem, is the following geometrical question asked by Geoffrey Colin Shephard (1964): if K and L are centrally symmetric convex bodies in n-dimensional Euclidean space such that whenever K and L are projected onto a hyperplane, the volume of the projection of K is smaller than the volume of the projection of L, then does it follow that the volume of K is smaller than that of L?In this case, "centrally symmetric" means that the reflection of K in the origin, −K, is a translate of K, and similarly for L. ".
- Shephard's_problem label "Shephard's problem".
- Shephard's_problem sameAs m.02qk7sg.
- Shephard's_problem sameAs Q7494442.
- Shephard's_problem sameAs Q7494442.
- Shephard's_problem wasDerivedFrom Shephard's_problem?oldid=572421208.
- Shephard's_problem isPrimaryTopicOf Shephard's_problem.