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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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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
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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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
The covering index of convex bodies
"... Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and longstanding problems. Swanepoel [Mathematika 52 (2005), 47–52] introduced the covering parameter of a convex body as a means of quantifying its covering properties. ..."
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Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and longstanding problems. Swanepoel [Mathematika 52 (2005), 47–52] introduced the covering parameter of a convex body as a means of quantifying its covering properties
Cylindrical Partitions of Convex Bodies
, 2005
"... A cylindrical partition of a convex body in R n is a partition of the body into subsets of smaller diameter, obtained by intersecting the body with a collection of mutually parallel convexbase cylinders. Convex bodies of constant width are characterized as those that do not admit a cylindrical pa ..."
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A cylindrical partition of a convex body in R n is a partition of the body into subsets of smaller diameter, obtained by intersecting the body with a collection of mutually parallel convexbase cylinders. Convex bodies of constant width are characterized as those that do not admit a cylindrical
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