Error of Asymptotic Formulae for Volume Approximation of Convex Bodies in

AbstractGiven r > 1, we consider convex bodies in En which contain a fixed unit ball, and whose extreme points are of distance at least r from the centre of the unit ball, and we investigate how well these convex bodies approximate the unit ball in terms of volume, surface area and mean width. As r tends to one, we prove asymptotic formulae for the error of the approximation, and provide good estimates on the involved constants depending on the dimension.

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Optimal Non-adaptive Approximation of Convex Bodies by Polytopes

Lecture Notes in Computational Science and Engineering - Numerical Geometry, Grid Generation and Scientific Computing ◽

10.1007/978-3-030-23436-2_11 ◽

2019 ◽

pp. 157-172

Author(s):

George K. Kamenev

Keyword(s):

Convex Bodies ◽

Adaptive Approximation ◽

Approximation Of Convex Bodies

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Fine approximation of convex bodies by polytopes

American Journal of Mathematics ◽

10.1353/ajm.2020.0018 ◽

2020 ◽

Vol 142 (3) ◽

pp. 809-820

Author(s):

Márton Naszódi ◽

Fedor Nazarov ◽

Dmitry Ryabogin

Keyword(s):

Convex Bodies ◽

Approximation Of Convex Bodies

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Asymptotic Formulae for the Lattice Point Enumerator

Canadian Journal of Mathematics ◽

10.4153/cjm-1999-012-9 ◽

1999 ◽

Vol 51 (2) ◽

pp. 225-249 ◽

Cited By ~ 2

Author(s):

U. Betke ◽

K. Böröczky

Keyword(s):

Convex Body ◽

Surface Area ◽

Positive Curvature ◽

Convex Bodies ◽

Mixed Volume ◽

Lattice Points ◽

Asymptotic Formulae ◽

General Convex ◽

Lattice Surface ◽

Small Deformations

AbstractLet M be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large λ the number of lattice points in λM is given by G(λM) = V(λM) + O(λd−1−ε(d)) for some positive ε(d). Here we give for general convex bodies the weaker estimatewhere SZd (M) denotes the lattice surface area of M. The term SZd is optimal for all convex bodies and o(λd−1) cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of M.Further we deal with families {Pλ} of convex bodies where the only condition is that the inradius tends to infinity. Here we havewhere the convex body K satisfies some simple condition, V(Pλ; K; 1) is some mixed volume and S(Pλ) is the surface area of Pλ.

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