منابع مشابه
Convex Polytopes
The study of convex polytopes in Euclidean space of two and three dimensions is one of the oldest branches of mathematics. Yet many of the more interesting properties of polytopes have been discovered comparatively recently, and are still unknown to the majority of mathematicians. In this paper we shall survey the subject, mentioning some of the most recent results, and stating the more importa...
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Given a convex body C ⊂ Rd containing the origin in its interior and a real number τ > 1 we seek to construct a polytope P ⊂ C with as few vertices as possible such that C ⊂ τP . Our construction is nearly optimal for a wide range of d and τ . In particular, we prove that if C = −C then for any 1 > > 0 and τ = 1 + one can choose P having roughly −d/2 vertices and for τ = √ d one can choose P ha...
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We prove that for every convex body K with the center of mass at the origin and every ε ∈ ( 0, 12 ) , there exists a convex polytope P with at most eO(d)ε− d−1 2 vertices such that (1− ε)K ⊂ P ⊂ K.
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Given a convex set and an interior point close to the boundary, we prove the existence of a supporting hyperplane whose distance to the point is controlled, in a dimensionally quantified way, by the thickness of the convex set in the orthogonal direction. This result has important applications in the regularity theory for Monge-Ampère type equations arising in optimal transportation.
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We discuss valuations on convex sets of oriented hyperplanes in R. For d = 2, we prove an analogue of Hadwiger’s characterization theorem for continuous, rigid motion invariant valuations.
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ژورنال
عنوان ژورنال: Mathematics
سال: 2019
ISSN: 2227-7390
DOI: 10.3390/math7050381