The convex hull of samples from self-similar distributions

1999 ◽  
Vol 31 (1) ◽  
pp. 34-47 ◽  
Author(s):  
Irene Hueter
Keyword(s):  
Convex Hull ◽  
Unit Disk ◽  
Analogous Result ◽  
Main Tool ◽  
Renewal Theorem ◽  
Similar Measure ◽  
Geometric Application ◽  
Self Similar

Let X1, X2,… be i.i.d. random points in ℝ2 with distribution ν, and let Nn denote the number of points spanning the convex hull of X1, X2,…,Xn. We obtain lim infn→∞E(Nn)n-1/3 ≥ γ1 and E(Nn) ≤ γ2n1/3(logn)2/3 for some positive constants γ1, γ2 and sufficiently large n under the assumption that ν is a certain self-similar measure on the unit disk. Our main tool consists in a geometric application of the renewal theorem. Exactly the same approach can be adopted to prove the analogous result in ℝd.

A Geometric Application of the “Shepherd's Principle”

1971 ◽  
Vol 64 (8) ◽  
pp. 756-758
Author(s):  
Gene Murrow
Keyword(s):  
Checkerboard Pattern ◽  
Standard Problem ◽  
Geometric Application

Most readers are probably familiar with the standard problem “How many squares of all size are contained in an n × n checkerboard pattern of n2 1 × 1 squares?”

scholarly journals Second-Order Elliptic Equation with Singularities

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Hichem Boughazi
Keyword(s):  
Riemannian Manifold ◽  
Elliptic Equation ◽  
Compact Riemannian Manifold ◽  
Second Order ◽  
Singular Equation ◽  
Order Elliptic Equation ◽  
Nontrivial Solutions ◽  
Geometric Application ◽  
Second Order Elliptic Equation

On the compact Riemannian manifold of dimension n≥5, we study the existence and regularity of nontrivial solutions for nonlinear second-order elliptic equation with singularities. At the end, we give a geometric application of the above singular equation.

scholarly journals Geometric application of a theorem on determinants

1873 ◽  
Vol 002 (1) ◽  
pp. 69-82
Author(s):  
F. J. Studnička
Keyword(s):  
Geometric Application

A Geometric Application of f(n)=n/(n+1)

1968 ◽  
Vol 41 (5) ◽  
pp. 259
Author(s):  
Francine Abeles
Keyword(s):  
Geometric Application

The convex hull of samples from self-similar distributions

1999 ◽  
Vol 31 (01) ◽  
pp. 34-47
Author(s):  
Irene Hueter
Keyword(s):  
Convex Hull ◽  
Unit Disk ◽  
Analogous Result ◽  
Main Tool ◽  
Renewal Theorem ◽  
Similar Measure ◽  
Geometric Application ◽  
Self Similar

Let X 1, X 2,… be i.i.d. random points in ℝ2 with distribution ν, and let N n denote the number of points spanning the convex hull of X 1, X 2,…,X n . We obtain lim inf n→∞ E (N n )n -1/3 ≥ γ1 and E (N n ) ≤ γ2 n 1/3(logn)2/3 for some positive constants γ1, γ2 and sufficiently large n under the assumption that ν is a certain self-similar measure on the unit disk. Our main tool consists in a geometric application of the renewal theorem. Exactly the same approach can be adopted to prove the analogous result in ℝ d .

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