Surface area deviation between smooth convex bodies and polytopes

AbstractWe provide general inequalities that compare the surface area S(K) of a convex body K in ℝn to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

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Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

Canadian Journal of Mathematics ◽

10.4153/cjm-2008-001-x ◽

2008 ◽

Vol 60 (1) ◽

pp. 3-32 ◽

Cited By ~ 1

Author(s):

Károly Böröczky ◽

Károly J. Böröczky ◽

Carsten Schütt ◽

Gergely Wintsche

Keyword(s):

Unit Ball ◽

Surface Area ◽

Convex Bodies ◽

Extreme Points ◽

Thin Shells ◽

Minimal Volume ◽

Asymptotic Formulae ◽

Mean Width

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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Continuity of the solution to the even Lp Minkowski problem for 0 < p < 1 in the plane

International Journal of Mathematics ◽

10.1142/s0129167x20501013 ◽

2020 ◽

Vol 31 (12) ◽

pp. 2050101

Author(s):

Hejun Wang ◽

Yusha Lv

Keyword(s):

Weak Convergence ◽

Surface Area ◽

Hausdorff Metric ◽

Convex Bodies ◽

Minkowski Problem

This paper concerns the continuity of the solution to the even [Formula: see text] Minkowski problem in the plane. When [Formula: see text], it is proved that the weak convergence of the even [Formula: see text] surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric. Moreover, the continuity of the solution to the even [Formula: see text] Minkowski problem with respect to [Formula: see text] is also obtained.

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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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A Class of Star-Shaped Bodies

Canadian Mathematical Bulletin ◽

10.4153/cmb-1959-023-6 ◽

1959 ◽

Vol 2 (3) ◽

pp. 175-180 ◽

Cited By ~ 5

Author(s):

Z.A. Melzak

Keyword(s):

Surface Area ◽

Lipschitz Condition ◽

Convex Bodies ◽

Nonempty Interior ◽

Empty Interior ◽

Selection Principle ◽

Positive Functions ◽

Bounded Sets ◽

Bounded Convex

The more important properties of the class κ of all bounded convex bodies in E3 with non-empty interior include: uniform approximability by polyhedra, existence of volume and surface area, and Blaschke's selection principle, [l ], [2 ]. In this note we define and consider a class ℋ of star-shaped bodies in E3, which enjoys many properties of κ, among them the above-mentioned ones, and is considerably larger. Roughly speaking, ℋ consists of closed bounded sets in E3 with nonempty interior, whose boundary is completely visible from every point of a set with non-empty interior. It turns out that ℋ is identifiable with the class of all real-valued positive functions on the sphere S3 which satisfy a Lipschitz condition.

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