The convex floating body.

Schütt, Carsten; Werner, Elisabeth M.

doi:10.7146/math.scand.a-12311

Cited by 164 publications

(227 citation statements)

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“…Theorem 2.3 [Schütt and Werner 1990]. For a convex body K containing the unit ball of a Euclidean space and p ∈ ∂ K , let R( p) ∈ [0, ∞) be the radius of the biggest ball contained in K and containing p. Then for all 0 < α < 1,…”

Section: Preliminaries On Convex Bodies and Hilbert Geometriesmentioning

confidence: 99%

Volume entropy of Hilbert geometries

Berck¹,

Bernig²,

Vernicos³

2010

Pacific J. Math.

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We show that among all plane Hilbert geometries, the hyperbolic plane has maximal volume entropy. More precisely, we show that the volume entropy is bounded above by 2/(3 − d) ≤ 1, where d is the Minkowski dimension of the extremal set of K , and we construct an explicit example of a plane Hilbert geometry with noninteger volume entropy. In arbitrary dimension, the hyperbolic space has maximal entropy among all Hilbert geometries satisfying some additional technical hypothesis. To achieve this result, we construct a new projective invariant of convex bodies, similar to the centroaffine area.

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Section: Preliminaries On Convex Bodies and Hilbert Geometriesmentioning

confidence: 99%

Volume entropy of Hilbert geometries

Berck¹,

Bernig²,

Vernicos³

2010

Pacific J. Math.

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“…Since the curvature κ is defined almost everywhere and Lebesgue integrable, the same definition can be used for general convex discs. This was pointed out by C. Schütt and E. Werner [18] for d-dimensional convex bodies and affine surface area. They showed that this way of defining affine surface area is equivalent to a definition given earlier by K. Leichtweiß [9], who was the first to consider affine surface area for general convex bodies in d-dimensional space.…”

Section: Introductionmentioning

confidence: 94%

A characterization of affine length and asymptotic approximation of convex discs

Ludwig

1999

Abh.Math.Semin.Univ.Hambg.

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“…is well-defined (see Schütt, Werner [32]). In addition a flag of a polytope P in R d is a sequence F 0 ⊂ .…”

Section: Related Problemsmentioning

confidence: 99%

Konvexgeometrie

Ball¹,

Goodey²,

Gruber³

2007

Oberwolfach Rep.

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Abstract. The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of other algorithms in computer science. High-dimensional geometry, both the discrete and convex branches of it, has experienced a striking series of developments in the past 5 years. Several examples were presented at this meeting, for example the work of Naor on the non-linear Dvoretzky theorem, that of Paouris on the distribution of the Euclidean norm on a convex domain and the results of Rudelson on the singular values of random matrices.

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The convex floating body.

Cited by 164 publications

References 0 publications

Volume entropy of Hilbert geometries

Volume entropy of Hilbert geometries

A characterization of affine length and asymptotic approximation of convex discs

Konvexgeometrie

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