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APPROXIMATION OF CONVEX BODIES BY CONVEX BODIES
"... Abstract. For the affine distance d(C, D) between two convex bodies C, D ⊂ R n, which reduces to the BanachMazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upperbounds are that F. John proved d(C, D) ≤ n 1 2 if o ..."
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Cited by 1 (0 self)
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Abstract. For the affine distance d(C, D) between two convex bodies C, D ⊂ R n, which reduces to the BanachMazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upperbounds are that F. John proved d(C, D) ≤ n 1 2
Concentration of mass on convex bodies,
 Geom. Funct. Anal.,
, 2006
"... Abstract We establish a sharp concentration of mass inequality for isotropic convex bodies: there exists an absolute constant c > 0 such that if K is an isotropic convex body in R n , then for every t 1, where LK denotes the isotropic constant. ..."
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Cited by 59 (4 self)
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Abstract We establish a sharp concentration of mass inequality for isotropic convex bodies: there exists an absolute constant c > 0 such that if K is an isotropic convex body in R n , then for every t 1, where LK denotes the isotropic constant.
Learning convex bodies is hard
"... We show that learning a convex body in Rd, given random samples from the body, requires 2Ω(√d/ɛ) samples. By learning a convex body we mean finding a set having at most ɛ relative symmetric difference with the input body. To prove the lower bound we construct a hard to learn family of convex bodies. ..."
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Cited by 7 (1 self)
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We show that learning a convex body in Rd, given random samples from the body, requires 2Ω(√d/ɛ) samples. By learning a convex body we mean finding a set having at most ɛ relative symmetric difference with the input body. To prove the lower bound we construct a hard to learn family of convex bodies
Order types of convex bodies
"... We prove a Hadwiger transversal type result, characterizing convex position on a family of noncrossing convex bodies in the plane. This theorem suggests a definition for the order type of a family of convex bodies, generalizing the usual definition of order type for point sets. This order type turn ..."
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Cited by 4 (2 self)
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We prove a Hadwiger transversal type result, characterizing convex position on a family of noncrossing convex bodies in the plane. This theorem suggests a definition for the order type of a family of convex bodies, generalizing the usual definition of order type for point sets. This order type
Moment inequalities and central limit properties of isotropic convex bodies
 Math. Zeitschr
, 2002
"... convex bodies ..."
On the Algebra of Intervals and Convex Bodies
 J. UCS
, 1998
"... Abstract: We introduce and study abstract algebraic systems generalizing the arithmetic systems of intervals and convex bodies involving Minkowski operations such as quasimodules and quasilinear systems. Embedding theorems are proved and computational rules for algebraic transformations are given. ..."
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Cited by 5 (0 self)
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Abstract: We introduce and study abstract algebraic systems generalizing the arithmetic systems of intervals and convex bodies involving Minkowski operations such as quasimodules and quasilinear systems. Embedding theorems are proved and computational rules for algebraic transformations are given.
THE COMPUTATIONAL COMPLEXITY OF CONVEX BODIES
, 2006
"... Abstract. We discuss how well a given convex body B in a real ddimensional vector space V can be approximated by a set X for which the membership question: “given an x ∈ V, does x belong to X? ” can be answered efficiently (in time polynomial in d). We discuss approximations of a convex body by an ..."
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Cited by 2 (0 self)
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Abstract. We discuss how well a given convex body B in a real ddimensional vector space V can be approximated by a set X for which the membership question: “given an x ∈ V, does x belong to X? ” can be answered efficiently (in time polynomial in d). We discuss approximations of a convex body
Learning Convex Bodies
"... � Can we try to learn the concepts under certain “natural ” distributions? � [GR09] : Convex bodies are hard to learn even under the uniform distribution � More specifically, there are convex bodies which force Ω every learning algorithm to draw at least ( d / ε) 2 samples from the uniform distribut ..."
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� Can we try to learn the concepts under certain “natural ” distributions? � [GR09] : Convex bodies are hard to learn even under the uniform distribution � More specifically, there are convex bodies which force Ω every learning algorithm to draw at least ( d / ε) 2 samples from the uniform
On the quermassintegrals of convex bodies
, 2013
"... The wellknown question for quermassintegrals is the following: For which values of i ∈N and every pair of convex bodies K and L, is it true that Wi(K + L) Wi+1(K + L) ..."
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The wellknown question for quermassintegrals is the following: For which values of i ∈N and every pair of convex bodies K and L, is it true that Wi(K + L) Wi+1(K + L)
Results 1  10
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83,322