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Elisabeth M. Werner - One of the best experts on this subject based on the ideXlab platform.
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Convex Floating Bodies of Equilibrium
arXiv: Metric Geometry, 2020Co-Authors: D.i. Florentin, Elisabeth M. Werner, Carsten Schuett, N. ZhangAbstract:We study a long standing open problem by Ulam, which is whether the Euclidean Ball is the unique body of uniform density which will float in equilibrium in any direction. We answer this problem in the class of origin symmetric n-dimensional convex bodies whose relative density to water is 1/2. For n=3, this result is due to Falconer.
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The Surface Area Deviation of the Euclidean Ball and a Polytope
Journal of Theoretical Probability, 2016Co-Authors: Steven D. Hoehner, Carsten Schütt, Elisabeth M. WernerAbstract:While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex bodies by arbitrarily positioned polytopes with a fixed number of vertices or facets in the symmetric surface area deviation.
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The Surface Area Deviation of the Euclidean Ball and a Polytope
arXiv: Probability, 2015Co-Authors: Steven D. Hoehner, Carsten Schuett, Elisabeth M. WernerAbstract:While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex bodies by arbitrarily positioned polytopes with a fixed number of vertices in the symmetric surface area deviation.
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Approximation of the Euclidean Ball by polytopes
Studia Mathematica, 2006Co-Authors: Monika Ludwig, Carsten Schütt, Elisabeth M. WernerAbstract:There is a constant c such that for every n ∈ N, there is a Nn so that for every N ≥ Nn there is a polytope P in Rn with N vertices and voln(B 24P ) ≤ c voln(B 2 )N − 2 n−1 where Bn 2 denotes the Euclidean unit Ball of dimension n. ∗partially supported by a grant from the National Science Foundation, from a Nato Collaborative Linkage Grant, and from a NSF Advance Opportunity Grant. 1991 Mathematics Subject Classification: 52 A 20
Carsten Schütt - One of the best experts on this subject based on the ideXlab platform.
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The Surface Area Deviation of the Euclidean Ball and a Polytope
Journal of Theoretical Probability, 2016Co-Authors: Steven D. Hoehner, Carsten Schütt, Elisabeth M. WernerAbstract:While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex bodies by arbitrarily positioned polytopes with a fixed number of vertices or facets in the symmetric surface area deviation.
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simplices in the Euclidean Ball
Canadian Mathematical Bulletin, 2012Co-Authors: Matthieu Fradelizi, Grigoris Paouris, Carsten SchüttAbstract:We establish some inequalities for the second moment 1 |K| ∫ K |x|2 dx of a convex body K under various assumptions on the position of K.
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Approximation of the Euclidean Ball by polytopes
Studia Mathematica, 2006Co-Authors: Monika Ludwig, Carsten Schütt, Elisabeth M. WernerAbstract:There is a constant c such that for every n ∈ N, there is a Nn so that for every N ≥ Nn there is a polytope P in Rn with N vertices and voln(B 24P ) ≤ c voln(B 2 )N − 2 n−1 where Bn 2 denotes the Euclidean unit Ball of dimension n. ∗partially supported by a grant from the National Science Foundation, from a Nato Collaborative Linkage Grant, and from a NSF Advance Opportunity Grant. 1991 Mathematics Subject Classification: 52 A 20
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A Simple Proof of an Estimate for the Approximation of the Euclidean Ball and the Delone Triangulation Numbers
Journal of Approximation Theory, 2000Co-Authors: Piotr Mankiewicz, Carsten SchüttAbstract:We give a simple proof of an estimate for the approximation of the Euclidean Ball by a polytope with a given number of vertices with respect to the volume of the symmetric difference metric and relatively precise estimate for the Delone triangulation numbers. We also study the same problem for a given number of n-1-dimensional faces.
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Umbrellas and Polytopal Approximation of the Euclidean Ball
Journal of Approximation Theory, 1997Co-Authors: Yehoram Gordon, Shlomo Reisner, Carsten SchüttAbstract:There are two positive, absolute constantsc1andc2so that the volume of the difference set of thed-dimensional Euclidean BallBd2and an inscribed polytope withnvertices is larger thanc1dvold(Bd2)n?2/(d?1)forn?(c2d)(d?1)/2.
Bo'az Klartag - One of the best experts on this subject based on the ideXlab platform.
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Rate of convergence of geometric symmetrizations
Geometrical and Functional Analysis GAFA, 2004Co-Authors: Bo'az KlartagAbstract:It is a classical fact, that given an arbitrary convex body \(K \subset \mathbb{R}^n ,\) there exists an appropriate sequence of Minkowski symmetrizations (or Steiner symmetrizations), that converges in Hausdorff metric to a Euclidean Ball. Here we provide quantitative estimates regarding this convergence, for both Minkowski and Steiner symmetrizations. Our estimates are polynomial in the dimension and in the logarithm of the desired distance to a Euclidean Ball, improving previously known exponential estimates. Inspired by a method of Diaconis [D], our technique involves spherical harmonics. We also make use of an earlier result by the author regarding “isomorphic Minkowski symmetrization”.
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5n Minkowski symmetrizations suffice to arrive at an approximate Euclidean Ball
arXiv: Functional Analysis, 2002Co-Authors: Bo'az KlartagAbstract:This paper proves that for every convex body in R^n there exist 5n-4 Minkowski symmetrizations, which transform the body into an approximate Euclidean Ball. This result complements the sharp c n log n upper estimate by J. Bourgain, J. Lindenstrauss and V.D. Milman, of the number of random Minkowski symmetrizations sufficient for approaching an approximate Euclidean Ball.
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$5n$ Minkowski symmetrizations suffice to arrive at an approximate Euclidean Ball
The Annals of Mathematics, 2002Co-Authors: Bo'az KlartagAbstract:This paper proves that for every convex body in R n there exist 5n Minkowski symmetrizations which transform the body into an approximate Euclidean Ball. This result complements the sharp cnlogn upper estimate by J. Bourgain, J. Lindenstrauss and V.D. Milman, of the number of random Minkowski symmetrizations sufficient for approaching an approximate Euclidean Ball.
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Remarks on minkowski symmetrizations
Lecture Notes in Mathematics, 2000Co-Authors: Bo'az KlartagAbstract:Here we extend a result by J. Bourgain, J. Lindenstrauss, V.D. Milman on the number of random Minkowski symmetrizations needed to obtain an approximated Ball, if we start from an arbitrary convex body in ℝn. We also show that the number of “deterministic” symmetrizations needed to approximate an Euclidean Ball may be significantly smaller than the number of “random” ones.
Steven D. Hoehner - One of the best experts on this subject based on the ideXlab platform.
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The Surface Area Deviation of the Euclidean Ball and a Polytope
Journal of Theoretical Probability, 2016Co-Authors: Steven D. Hoehner, Carsten Schütt, Elisabeth M. WernerAbstract:While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex bodies by arbitrarily positioned polytopes with a fixed number of vertices or facets in the symmetric surface area deviation.
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The Surface Area Deviation of the Euclidean Ball and a Polytope
arXiv: Probability, 2015Co-Authors: Steven D. Hoehner, Carsten Schuett, Elisabeth M. WernerAbstract:While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex bodies by arbitrarily positioned polytopes with a fixed number of vertices in the symmetric surface area deviation.
Alexander Segal - One of the best experts on this subject based on the ideXlab platform.
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Minkowski symmetrizations of star shaped sets
Geometriae Dedicata, 2016Co-Authors: Dan I. Florentin, Alexander SegalAbstract:We provide sharp upper bounds for the number of symmetrizations required to transform a star shaped set in \({\mathbb {R}}^n\) arbitrarily close (in the Hausdorff metric) to the Euclidean Ball.
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Stability and Rate of Convergence of the Steiner Symmetrization
arXiv: Metric Geometry, 2015Co-Authors: Dan I. Florentin, Alexander SegalAbstract:We present a direct analytic method towards an estimate for the rate of convergence (to the Euclidean Ball) of Steiner symmetrizations. To this end we present a modified version of a known stability property of the Steiner symmetrization.
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On Convergence of Blaschke and Minkowski Symmetrization Through Stability Results
Lecture Notes in Mathematics, 2014Co-Authors: Alexander SegalAbstract:We show how existing results of stability for Brunn-Minkowski and related inequalities imply results regarding rate of convergences of Minkowski and Blaschke symmetrization processes to the Euclidean Ball. To be more precise, the results imply that the amount of symmetrizations needed to approach the Euclidean Ball within some distance e, a polynomial number of symmetrizations (in the dimension and \(\frac{1} {\epsilon }\)) suffice.
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Isomorphic Steiner symmetrization of p-convex sets
Geometriae Dedicata, 2013Co-Authors: Alexander SegalAbstract:In this paper we show that given a \(p\)-convex set \(K \subset \mathbb{R }^n\), there exist \(5n\) Steiner symmetrizations that transform it into an isomorphic Euclidean Ball. That is, if \(|K| = |D_n| = \kappa _n\), we may symmetrize it, using \(5n\) Steiner symmetrizations, into a set \(K'\) such that \(c_p D_n \subset K' \subset C_p D_n\), where \(c_p\) and \(C_p\) are constants dependent on \(p\) only.