On the mean number of normals through a point in the interior of a convex body

1995 ◽  
Vol 55 (3) ◽  
pp. 319-340 ◽  
Author(s):  
Daniel Hug
Keyword(s):  
Convex Body ◽  
The Mean

scholarly journals Complexity Analysis of a Sampling-Based Interior Point Method for Convex Optimization

2021 ◽  
Author(s):  
Riley Badenbroek ◽  
Etienne de Klerk
Keyword(s):  
Convex Body ◽  
Interior Point ◽  
Interior Point Method ◽  
Complexity Analysis ◽  
Point Method ◽  
Mixing Times ◽  
Approximation Quality ◽  
Gibbs Distributions ◽  
Hit And Run ◽  
The Mean

We develop a short-step interior point method to optimize a linear function over a convex body assuming that one only knows a membership oracle for this body. The approach is based a sketch of a universal interior point method using the so-called entropic barrier. It is well known that the gradient and Hessian of the entropic barrier can be approximated by sampling from Boltzmann-Gibbs distributions and the entropic barrier was shown to be self-concordant. The analysis of our algorithm uses properties of the entropic barrier, mixing times for hit-and-run random walks, approximation quality guarantees for the mean and covariance of a log-concave distribution, and results on inexact Newton-type methods.

scholarly journals The Mean Width of Circumscribed Random Polytopes

2010 ◽  
Vol 53 (4) ◽  
pp. 614-628 ◽  
Author(s):  
Károly J. Böröczky ◽  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Asymptotic Formula ◽  
Uniform Distribution ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Simplicial Polytope ◽  
Mean Width ◽  
The Mean ◽  
The Difference ◽  
Precise Asymptotic

AbstractFor a given convex body K in ℝd, a random polytope K(n) is defined (essentially) as the intersection of n independent closed halfspaces containing K and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of K(n) and K as n tends to infinity. For a simplicial polytope P, a precise asymptotic formula for the difference of the mean widths of P(n) and P is obtained.

scholarly journals Interaction of Poisson hyperplane processes and convex bodies

2019 ◽  
Vol 56 (4) ◽  
pp. 1020-1032 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Random Polytope ◽  
Mean Width ◽  
The Mean

AbstractGiven a stationary and isotropic Poisson hyperplane process and a convex body K in ${\mathbb R}^d$ , we consider the random polytope defined by the intersection of all closed half-spaces containing K that are bounded by hyperplanes of the process not intersecting K. We investigate how well the expected mean width of this random polytope approximates the mean width of K if the intensity of the hyperplane process tends to infinity.

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