Improved upper bounds for approximation by zonotopes

A zonotope in R n is a special type of a convex polytope; it is defined as a Minkowski sum of finitely many segments. Tha t is, a zonotope is a set in R n of the form {Xl +x2 +... +Xm : XlCI1,...',XmEIm}, where I1,...,Im are segments in R n. A convex body tha t can be approximated by zonotopes arbitrari ly closely is called a zonoid. Several authors have recently studied the following question: what is the minimum number, N, of summands of a zonotope needed to approximate a given zonoid Z in R ~ with error at most ~ (this means tha t ZCAC(I+~)Z , where A is the approximating zonotope, and we assume tha t the center of symmet ry of Z is the origin). Here we consider the dimension n fixed, and we investigate the dependence of N on ~ (we assume tha t n~>3, as the case n = 2 is simple---see [4]).

[1]  P. McMullen,et al.  Estimating the Sizes of Convex Bodies from Projections , 1983 .

[2]  J. Beck.  Some upper bounds in the theory of irregularities of distribution , 1984 .

[3]  J. Spencer.  Six standard deviations suffice , 1985 .

[4]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[5]  Bernard Chazelle,et al.  A deterministic view of random sampling and its use in geometry , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[6]  J. Lindenstrauss,et al.  Distribution of points on spheres and approximation by zonotopes , 1988 .

[7]  J. Lindenstrauss,et al.  Approximation of zonoids by zonotopes , 1989 .

[8]  Bernard Chazelle,et al.  Quasi-optimal range searching in spaces of finite VC-dimension , 1989, Discret. Comput. Geom..

[9]  J. Linhart,et al.  Approximation of a ball by zonotopes using uniform distribution on the sphere , 1989 .

[10]  Jirí Matousek,et al.  Cutting hyperplane arrangements , 1990, SCG '90.

[11]  Jirí Matousek,et al.  Efficient partition trees , 1991, SCG '91.

[12]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[13]  Gerold Wagner.  On a new method for constructing good point sets on spheres , 1993, Discret. Comput. Geom..

[14]  Bernard Chazelle,et al.  Cutting hyperplanes for divide-and-conquer , 1993, Discret. Comput. Geom..

[15]  Jean Bourgain,et al.  Approximating the ball by a minkowski sum of segments with equal length , 1993, Discret. Comput. Geom..

[16]  Jirí Matousek,et al.  Tight upper bounds for the discrepancy of half-spaces , 1995, Discret. Comput. Geom..

[17]  Jiří Matoušek,et al.  Discrepancy in arithmetic progressions , 1996 .