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Elisabeth M Werner - One of the best experts on this subject based on the ideXlab platform.
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constrained Convex bodies with extremal affine surface areas
arXiv: Functional Analysis, 2019Co-Authors: Ohad Giladi, Carsten Schütt, Elisabeth M Werner, H HuangAbstract:Given a Convex Body K in R^n and p in R, we introduce and study the extremal inner and outer affine surface areas IS_p(K) = sup_{K'\subseteq K} (as_p(K') ) and os_p(K)=inf_{K'\supseteq K} (as_p(K') ), where as_p(K') denotes the L_p-affine surface area of K', and the supremum is taken over all Convex subsets of K and the infimum over all Convex compact subsets containing K. The Convex Body that realizes IS_1(K) in dimension 2 was determined by Barany. He also showed that this Body is the limit shape of lattice polytopes in K. In higher dimensions no results are known about the extremal bodies. We use a thin shell estimate of Guedon and Milman and the Lowner ellipsoid to give asymptotic estimates on the size of IS_p(K) and os_p(K). Surprisingly, both quantities are proportional to a power of volume.
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the spherical Convex floating Body
Advances in Mathematics, 2016Co-Authors: Florian Besau, Elisabeth M WernerAbstract:Abstract For a Convex Body on the Euclidean unit sphere the spherical Convex floating Body is introduced. The asymptotic behavior of the volume difference of a spherical Convex Body and its spherical floating Body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area turns out to be a spherical analogue to the classical affine surface area from affine differential geometry. Several properties of the floating area are established.
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renyi divergence and lp affine surface area for Convex bodies
Advances in Mathematics, 2012Co-Authors: Elisabeth M WernerAbstract:Abstract We show that the fundamental objects of the L p -Brunn–Minkowski theory, namely the L p -affine surface areas for a Convex Body, are closely related to information theory: they are exponentials of Renyi divergences of the cone measures of a Convex Body and its polar. We give geometric interpretations for all Renyi divergences D α , not just for the previously treated special case of relative entropy which is the case α = 1 . Now, no symmetry assumptions are needed and, if at all, only very weak regularity assumptions are required. Previously, the relative entropies appeared only after performing second order expansions of certain expressions. Now already first order expansions make them appear. Thus, in the new approach we detect “faster” details about the boundary of a Convex Body.
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polytopes with vertices chosen randomly from the boundary of a Convex Body
2003Co-Authors: Carsten Schütt, Elisabeth M WernerAbstract:Let K be a Convex Body in \(\mathbb{R}^n\) and let \(f : \partial K \to \mathbb{R}_ + \) be a continuous, positive function with \(\int_{\partial K} f(x )d\mu_{\partial K} (x ) = 1\) where \(\mu_{\partial K}\) is the surface measure on \(\partial K\). Let \(\mathbb{P}_f\) be the probability measure on \(\partial K\) given by \({\rm d}\mathbb{P}_f(x ) = f (x ){\rm d} \mu_{\partial K} (x )\). Let \(\kappa\) be the (generalized) Gaus-Kronecker curvature and \(\mathbb{E}(f,N )\) the expected volume of the Convex hull of N points chosen randomly on \(\partial K\) with respect to \(\mathbb{P}_f\). Then, under some regularity conditions on the boundary of K $$ \lim_{ N\to \infty} \frac{{\rm vol}_n (K ) -\mathbb{E} (f,N )}{\left(\frac{1}{N}\right)^{\frac{2}{n-1}}} = c_n\int_{\partial K}\frac{\kappa(x)^{\frac{1}{n-1}}}{f(x)^{\frac{2}{n-1}}} {\rm d}\mu_{\partial K}(x),$$ where c n is a constant depending on the dimension n only.
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the santalo regions of a Convex Body
Transactions of the American Mathematical Society, 1998Co-Authors: Mathieu Meyer, Elisabeth M WernerAbstract:Motivated by the Blaschke-Santalo inequality, we define for a Convex Body K in R and for t ∈ R the Santalo-regions S(K,t) of K. We investigate properties of these sets and relate them to a concept of Affine Differential Geometry, the affine surface area of K. Let K be a Convex Body in R. For x ∈ int(K), the interior of K, let K be the polar Body of K with respect to x. It is well known that there exists a unique x0 ∈ int(K) such that the product of the volumes |K||K0 | is minimal (see for instance [Sch]). This unique x0 is called the Santalo-point of K. Moreover the Blaschke-Santalo inequality says that |K||K0 | ≤ v n (where vn denotes the volume of the n-dimensional Euclidean unit ball B(0, 1)) with equality if and only if K is an ellipsoid. For t ∈ R we consider here the sets S(K, t) = {x ∈ K : |K||K | v2 n ≤ t}. Following E. Lutwak, we call S(K, t) a Santalo-region of K. Observe that it follows from the Blaschke-Santalo inequality that the Santalopoint x0 ∈ S(K, 1) and that S(K, 1) = {x0} if and only if K is an ellipsoid. Thus S(K, t) has non-empty interior for some t < 1 if and only if K is not an ellipsoid. In the first part of this paper we show some properties of S(K, t) and give estimates on the “size” of S(K, t). This question was asked by E. Lutwak. ∗the paper was written while both authors stayed at MSRI †supported by a grant from the National Science Foundation. MSC classification 52
Partha Niyogi - One of the best experts on this subject based on the ideXlab platform.
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heat flow and a faster algorithm to compute the surface area of a Convex Body
Random Structures and Algorithms, 2013Co-Authors: Mikhail Belkin, H Narayanan, Partha NiyogiAbstract:We draw on the observation that the amount of heat diffusing outside of a heated Body in a short period of time is proportional to its surface area, to design a simple algorithm for approximating the surface area of a Convex Body given by a membership oracle. Our method has a complexity of O*(n4), where n is the dimension, compared to O*(n8) for the previous best algorithm. We show that our complexity cannot be improved given the current state-of-the-art in volume estimation. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 407–428, 2013
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heat flow and a faster algorithm to compute the surface area of a Convex Body
Foundations of Computer Science, 2006Co-Authors: Mikhail Belkin, H Narayanan, Partha NiyogiAbstract:We draw on the observation that the amount of heat diffusing outside of a heated Body in a short period of time is proportional to its surface area, to design a simple algorithm for approximating the surface area of a Convex Body given by a membership oracle. Our method has a complexity of O*(n^4), where n is the dimension, compared to O*(n^8.5) for the previous best algorithm. We show that our complexity cannot be improved given the current state-of-the-art in volume estimation.
Alexander Volberg - One of the best experts on this subject based on the ideXlab platform.
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Convex Body domination and weighted estimates with matrix weights
Advances in Mathematics, 2017Co-Authors: Fedor Nazarov, Stefanie Petermichl, Sergei Treil, Alexander VolbergAbstract:Abstract We introduce the so called Convex Body valued sparse operators, which generalize the notion of sparse operators to the case of spaces of vector valued functions. We prove that Calderon–Zygmund operators as well as Haar shifts and paraproducts can be dominated by such operators. By estimating sparse operators we obtain weighted estimates with matrix weights. We get two weight A 2 – A ∞ estimates, that in the one weight case give us the estimate ‖ T ‖ L 2 ( W ) → L 2 ( W ) ≤ C [ W ] A 2 1 / 2 [ W ] A ∞ ≤ C [ W ] A 2 3 / 2 where T is either Calderon–Zygmund operator (with modulus of continuity satisfying the Dini condition), or a Haar shift or a paraproduct.
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Convex Body domination and weighted estimates with matrix weights
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Fedor Nazarov, Stefanie Petermichl, Sergei Treil, Alexander VolbergAbstract:We introduce the so called Convex Body valued sparse operators, which generalize the notion of sparse operators to the case of spaces of vector valued functions. We prove that Calderon--Zygmund operators as well as Haar shifts and paraproducts can be dominated by such operators. By estimating sparse operators we obtain weighted estimates with matrix weights. We get two weight $A_2$-$A_\infty$ estimates, that in the one weight case give us the estimate $$ \|T\|_{L^2(W)\to L^2 (W)} \le C [W]_{A_2}^{1/2} [W]_{A_\infty} \le C[W]_{A_2}^{3/2} $$ where $T$ is either a Calderon--Zygmund operator (with modulus of continuity satisfying the Dini condition), or a Haar shift or a paraproduct.
Carsten Schütt - One of the best experts on this subject based on the ideXlab platform.
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constrained Convex bodies with extremal affine surface areas
arXiv: Functional Analysis, 2019Co-Authors: Ohad Giladi, Carsten Schütt, Elisabeth M Werner, H HuangAbstract:Given a Convex Body K in R^n and p in R, we introduce and study the extremal inner and outer affine surface areas IS_p(K) = sup_{K'\subseteq K} (as_p(K') ) and os_p(K)=inf_{K'\supseteq K} (as_p(K') ), where as_p(K') denotes the L_p-affine surface area of K', and the supremum is taken over all Convex subsets of K and the infimum over all Convex compact subsets containing K. The Convex Body that realizes IS_1(K) in dimension 2 was determined by Barany. He also showed that this Body is the limit shape of lattice polytopes in K. In higher dimensions no results are known about the extremal bodies. We use a thin shell estimate of Guedon and Milman and the Lowner ellipsoid to give asymptotic estimates on the size of IS_p(K) and os_p(K). Surprisingly, both quantities are proportional to a power of volume.
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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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polytopes with vertices chosen randomly from the boundary of a Convex Body
2003Co-Authors: Carsten Schütt, Elisabeth M WernerAbstract:Let K be a Convex Body in \(\mathbb{R}^n\) and let \(f : \partial K \to \mathbb{R}_ + \) be a continuous, positive function with \(\int_{\partial K} f(x )d\mu_{\partial K} (x ) = 1\) where \(\mu_{\partial K}\) is the surface measure on \(\partial K\). Let \(\mathbb{P}_f\) be the probability measure on \(\partial K\) given by \({\rm d}\mathbb{P}_f(x ) = f (x ){\rm d} \mu_{\partial K} (x )\). Let \(\kappa\) be the (generalized) Gaus-Kronecker curvature and \(\mathbb{E}(f,N )\) the expected volume of the Convex hull of N points chosen randomly on \(\partial K\) with respect to \(\mathbb{P}_f\). Then, under some regularity conditions on the boundary of K $$ \lim_{ N\to \infty} \frac{{\rm vol}_n (K ) -\mathbb{E} (f,N )}{\left(\frac{1}{N}\right)^{\frac{2}{n-1}}} = c_n\int_{\partial K}\frac{\kappa(x)^{\frac{1}{n-1}}}{f(x)^{\frac{2}{n-1}}} {\rm d}\mu_{\partial K}(x),$$ where c n is a constant depending on the dimension n only.
Matthieu Fradelizi - One of the best experts on this subject based on the ideXlab platform.
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on the volume of sections of a Convex Body by cones
Proceedings of the American Mathematical Society, 2017Co-Authors: Matthieu Fradelizi, Mathieu Meyer, Vlad YaskinAbstract:Let $K$ be a Convex Body in $\mathbb R^n$. We prove that in small codimensions, the sections of a Convex Body through the centroid are quite symmetric with respect to volume. As a consequence of our estimates we give a positive answer to a problem posed by M. Meyer and S. Reisner regarding Convex intersection bodies.
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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.
