Approximation of the sphere by polytopes having few vertices

Bárány, I.; Füredi, Z.

doi:10.1090/S0002-9939-1988-0928998-8

Approximation of the sphere by polytopes having few vertices
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by I. Bárány and Z. Füredi PDF

Proc. Amer. Math. Soc. 102 (1988), 651-659 Request permission

Abstract:

How well can a polytope with $n$ vertices approximate the unit ball ${B^d}$ of the $d$-dimensional Euclidean space? The answer is quite well known when $d$ is fixed and $n$ tends to infinity. In this paper the same question is answered when $n$ is a function of $d$ (a polynomial in $d$, say) and $d$ tends to infinity. Some applications of the results are also indicated.

References

Imre Bárány and Zoltán Füredi, Computing the volume is difficult, Discrete Comput. Geom. 2 (1987), no. 4, 319–326. MR 911186, DOI 10.1007/BF02187886

On the convex hull of random points

C. Buchta, J. Müller, and R. F. Tichy, Stochastical approximation of convex bodies, Math. Ann. 271 (1985), no. 2, 225–235. MR 783553, DOI 10.1007/BF01455988

Helly’s theorem and its relatives

R. M. Dudley, Metric entropy of some classes of sets with differentiable boundaries, J. Approximation Theory 10 (1974), 227–236. MR 358168, DOI 10.1016/0021-9045(74)90120-8
G. Elekes, A geometric inequality and the complexity of computing volume, Discrete Comput. Geom. 1 (1986), no. 4, 289–292. MR 866364, DOI 10.1007/BF02187701
L. Fejes Tóth, Regular figures, A Pergamon Press Book, The Macmillan Company, New York, 1964. MR 0165423
Peter M. Gruber, Approximation of convex bodies, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 131–162. MR 731110
Peter M. Gruber and Petar Kenderov, Approximation of convex bodies by polytopes, Rend. Circ. Mat. Palermo (2) 31 (1982), no. 2, 195–225. MR 670396, DOI 10.1007/BF02844354

An algorithmic theory of numbers, graphs and convexity

A. M. Macbeath, An extremal property of the hypersphere, Proc. Cambridge Philos. Soc. 47 (1951), 245–247. MR 39292, DOI 10.1017/s0305004100026542
Rolf Schneider, Zur optimalen Approximation konvexer Hyperflächen durch Polyeder, Math. Ann. 256 (1981), no. 3, 289–301 (German). MR 626950, DOI 10.1007/BF01679698
Rolf Schneider and John André Wieacker, Approximation of convex bodies by polytopes, Bull. London Math. Soc. 13 (1981), no. 2, 149–156. MR 608101, DOI 10.1112/blms/13.2.149