Estimates for certain Orlicz–Sobolev capacities of an Euclidean ball

2010 ◽  
Vol 72 (11) ◽  
pp. 4316-4330 ◽  
Author(s):  
Jani Joensuu
Keyword(s):  
Euclidean Ball

scholarly journals Robin conditions on the Euclidean ball

1996 ◽  
Vol 13 (4) ◽  
pp. 585-610 ◽  
Author(s):  
J S Dowker
Keyword(s):  
Euclidean Ball

scholarly journals APPROXIMATING SMALLEST ENCLOSING BALLS WITH APPLICATIONS TO MACHINE LEARNING

2009 ◽  
Vol 19 (05) ◽  
pp. 389-414 ◽  
Author(s):  
FRANK NIELSEN ◽  
RICHARD NOCK
Keyword(s):  
Machine Learning ◽  
Hybrid Algorithms ◽  
Decision Problems ◽  
Dual Method ◽  
Euclidean Ball ◽  
Prior Work ◽  
Euclidean Spaces ◽  
Information Theoretic ◽  
Primal Dual ◽  
Covering Set

In this paper, we first survey prior work for computing exactly or approximately the smallest enclosing balls of point or ball sets in Euclidean spaces. We classify previous work into three categories: (1) purely combinatorial, (2) purely numerical, and (3) recent mixed hybrid algorithms based on coresets. We then describe two novel tailored algorithms for computing arbitrary close approximations of the smallest enclosing Euclidean ball. These deterministic heuristics are based on solving relaxed decision problems using a primal-dual method. The primal-dual method is interpreted geometrically as solving for a minimum covering set, or dually as seeking for a minimum piercing set. Finally, we present some applications in machine learning of the exact and approximate smallest enclosing ball procedure, and discuss about its extension to non-Euclidean information-theoretic spaces.

Optimal Estimates for Steklov Eigenvalue Gaps and Ratios on Warped Product Manifolds

2019 ◽  
Author(s):  
Changwei Xiong
Keyword(s):  
Riemannian Manifold ◽  
Warped Product ◽  
Eigenvalue Problems ◽  
Euclidean Ball ◽  
Convex Boundary ◽  
Steklov Eigenvalue ◽  
Optimal Estimates ◽  
Steklov Eigenvalue Problem ◽  
Steklov Eigenvalue Problems ◽  
Optimal Lower Bound

Abstract We consider an $n$-dimensional smooth Riemannian manifold $M^n=[0,R)\times \mathbb{S}^{n-1}$ endowed with a warped product metric $g=dr^2+h^2(r)g_{\mathbb{S}^{n-1}}$ and diffeomorphic to a Euclidean ball. Suppose that $M$ has strictly convex boundary. First, for the classical Steklov eigenvalue problem, we derive an optimal lower (upper, respectively) bound for its eigenvalue gaps in terms of $h^{\prime}(R)/h(R)$ when $n\geq 2$ and $Ric_g\geq 0$ ($\leq 0$, respectively). Second, in the same spirit, for two 4th-order Steklov eigenvalue problems studied by Kuttler and Sigillito in 1968, we deduce an optimal lower bound for their eigenvalue gaps in terms of either $h^{\prime}(R)/h^3(R)$ or $h^{\prime}(R)/h(R)$ when $n=2$ and the Gaussian curvature is nonnegative. We also consider optimal estimates on the eigenvalue ratios for these eigenvalue problems.

On CMC free-boundary stable hypersurfaces in a Euclidean ball

2018 ◽  
Vol 372 (1-2) ◽  
pp. 179-187 ◽  
Author(s):  
Ezequiel Barbosa
Keyword(s):  
Free Boundary ◽  
Euclidean Ball

scholarly journals Simplices in the Euclidean Ball

2012 ◽  
Vol 55 (3) ◽  
pp. 498-508 ◽  
Author(s):  
Matthieu Fradelizi ◽  
Grigoris Paouris ◽  
Carsten Schütt
Keyword(s):  
Convex Body ◽  
Euclidean Ball ◽  
Second Moment ◽  
Image Position

AbstractWe establish some inequalities for the second momentof a convex body K under various assumptions on the position of K.

Linear Interpolation on a Euclidean Ball in ℝn

2020 ◽  
Vol 54 (7) ◽  
pp. 601-614
Author(s):  
M. V. Nevskii ◽  
A. Yu. Ukhalov
Keyword(s):  
Linear Interpolation ◽  
Euclidean Ball

scholarly journals Asymptotic shape in a continuum growth model

2003 ◽  
Vol 35 (02) ◽  
pp. 303-318 ◽  
Author(s):  
Maria Deijfen
Keyword(s):  
Growth Model ◽  
Expected Value ◽  
The State ◽  
Rotational Invariance ◽  
Euclidean Ball ◽  
Bounded Support ◽  
Asymptotic Shape

A continuum growth model is introduced. The state at time t, S t , is a subset of ℝ d and consists of a connected union of randomly sized Euclidean balls, which emerge from outbursts at their centre points. An outburst occurs somewhere in S t after an exponentially distributed time with expected value |S t |-1 and the location of the outburst is uniformly distributed over S t . The main result is that, if the distribution of the radii of the outburst balls has bounded support, then S t grows linearly and S t /t has a nonrandom shape as t → ∞. Due to rotational invariance the asymptotic shape must be a Euclidean ball.

scholarly journals On the Local Convexity of Intersection Bodies of Revolution

2015 ◽  
Vol 67 (1) ◽  
pp. 3-27
Author(s):  
M. Angeles Alfonseca ◽  
Jaegil Kim
Keyword(s):  
Convex Body ◽  
Convex Geometry ◽  
Euclidean Ball ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Body Of Revolution ◽  
Local Convexity ◽  
Bodies Of Revolution ◽  
Intersection Body ◽  
Intersection Bodies

AbstractOne of the fundamental results in convex geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach–Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

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