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- In mathematics, Shephard's problem, is the following geometrical question asked by Geoffrey Colin Shephard in 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 implication Vk(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 and the notion of projection bodies of convex bodies. (en)
- Задача Шепарда — вопрос выпуклой геометрии о сравнении объёмов двух симметричных выпуклых тел при условии, что в любом направлении площадь проекции первого не превосходит площади проекции второго. Вопрос был сформулирован в 1964 году. Ответ на этот вопрос — «да» в размерности 2 и «нет» в размерности 3 и выше.Последнее было доказано независимо и Шнайдером в 1967 году. (ru)
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- Задача Шепарда — вопрос выпуклой геометрии о сравнении объёмов двух симметричных выпуклых тел при условии, что в любом направлении площадь проекции первого не превосходит площади проекции второго. Вопрос был сформулирован в 1964 году. Ответ на этот вопрос — «да» в размерности 2 и «нет» в размерности 3 и выше.Последнее было доказано независимо и Шнайдером в 1967 году. (ru)
- In mathematics, Shephard's problem, is the following geometrical question asked by Geoffrey Colin Shephard in 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? Vk(πk(K)) is sometimes known as the brightness of K and the function Vk o πk as a (k-dimensional) brightness function. (en)
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- Shephard's problem (en)
- Задача Шепарда (ru)
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