Approximation of convex bodies

Gruber, Peter M.

doi:10.1007/978-3-0348-5858-8_7

Cited by 98 publications

(55 citation statements)

References 80 publications

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“…1,2,5,6,7,20 Next we want to indicate how the concepts of floating body respectively illumination body and affine surface area are related. Notice that the above two theorems provide a geometric interpretation of the affine surface area in terms of volume differences of a convex body and its floating body respectively of its illumination body.…”

Section: Proofmentioning

confidence: 99%

Floating Bodies and Illumination Bodies

Werner

2006

Integral Geometry and Convexity

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Section: Proofmentioning

confidence: 99%

Floating Bodies and Illumination Bodies

Werner

2006

Integral Geometry and Convexity

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“…Given a convex curve γ , one wants to approximate it by circumscribed polygons using area as the distance between the curve and an approximating polygon; this problem makes sense in the Euclidean, hyperbolic and spherical geometries. Formula (2) provides an asymptotic expansion for the distance between γ and its best approximating n-gon; in the Euclidean case, the term a 1 was found by Fejes Toth [35,36] (see [20] for a complete proof, [18] for the value of the term a 2 and [7,8] for surveys on approximating convex bodies by polytops). The approach via interpolating Hamiltonians provides a novel view point in the approximation theory of smooth convex curves by polygons.…”

Section: Remarkmentioning

confidence: 99%

Dual billiards in the hyperbolic plane*

Tabachnikov

2002

Nonlinearity

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We study an area preserving map of the exterior of a smooth convex curve in the hyperbolic plane, defined by a natural geometrical construction and called the dual billiard map. We consider two problems: stability and the area spectrum. The dual billiard map is called stable if all its orbits are bounded. We show that both stable and unstable behaviours may occur. If the map at infinity has a hyperbolic periodic orbit, then the dual billiard map has orbits escaping to infinity. On the other extreme, if the map at infinity is smoothly conjugated to a Diophantine irrational rotation of the circle, then the dual billiard map is stable.The area spectrum is the set of extremal areas of n-gons, circumscribed about the dual billiard curve; this is to the dual billiard what the length spectrum is to the usual, inner, one. We show that the area spectrum has an asymptotic expansion in even negative powers of n as n → ∞. The first coefficient of this expansion is the area of the dual billiard curve, and the next is, up to a constant, the cubed integral of the cube root of its curvature. We describe the curves that are relative extrema of these two functionals and show that they are the trajectories of the pseudospherical pendulum with various gravity directions.

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“…In the case of the symmetric difference metric and inscribed and circumscribed polytopes these formulae were derived by P. M. Gruber in [5], [6], and [8] for any dimension; see also [10]. For detailed information see the surveys [4] and [7].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic approximation of convex curves

Ludwig

1994

Arch. Math

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Abstract. L. Fejes Tóth gave asymptotic formulae as n → ∞ for the distance between a smooth convex disc and its best approximating inscribed or circumscribed polygons with at most n vertices, where the distance is in the sense of the symmetric difference metric. In this paper these formulae are extended by specifying the second terms of the asymptotic expansions. Tools are from affine differential geometry.

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Approximation of convex bodies

Cited by 98 publications

References 80 publications

Floating Bodies and Illumination Bodies

Floating Bodies and Illumination Bodies

Dual billiards in the hyperbolic plane*

Asymptotic approximation of convex curves

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