On the average distance property of compact connected metric spaces

1983 ◽  
Vol 40 (1) ◽  
pp. 459-463 ◽  
Author(s):  
Sidney A. Morris ◽  
Peter Nickolas
Keyword(s):  
Metric Spaces ◽  
Average Distance ◽  
Distance Property ◽  
Average Distance Property

scholarly journals On the average distance property in finite dimensional real Banach spaces

1995 ◽  
Vol 51 (1) ◽  
pp. 87-101 ◽  
Author(s):  
Reinhard Wolf
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Positive Real Number ◽  
Real Banach Space ◽  
Average Distance ◽  
Positive Real ◽  
Finite Dimensional ◽  
Distance Property ◽  
Average Distance Property ◽  
Image Position

The average distance Theorem of Gross implies that for each N-dimensional real Banach space E (N ≥ 2) there is a unique positive real number r(E) with the following property: for each positive integer n and for all (not necessarily distinct) x1, x2, …, xn, in E with ‖x1‖ = ‖x2‖ = … = ‖xn‖ = 1, there exists an x in E with ‖x‖ = 1 such that.In this paper we prove that if E has a 1-unconditional basis then r(E)≤2−(l/N) and equality holds if and only if E is isometrically isomorphic to Rn equipped with the usual 1-norm.

scholarly journals Average distances in compact connected spaces

1982 ◽  
Vol 26 (3) ◽  
pp. 331-342 ◽  
Author(s):  
David Yost
Keyword(s):  
Convex Sets ◽  
Average Distance ◽  
Compact Convex ◽  
Topological Spaces ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Distance Property ◽  
Compact Convex Set ◽  
Average Distance Property ◽  
Image Position

We give a simple proof of the fact that compact, connected topological spaces have the “average distance property”. For a metric space (X, d), this asserts the existence of a unique number a = a(X) such that, given finitely many points x1, …, xn ∈ X, then there is some y ∈ X withWe examine the possible values of a(X) , for subsets of finite dimensional normed spaces. For example, if diam(X) denotes the diameter of some compact, convex set in a euclidean space, then a(X) ≤ diam(X)/√2 . On the other hand, a(X)/diam(X) can be arbitrarily close to 1 , for non-convex sets in euclidean spaces of sufficiently large dimension.

scholarly journals The average distance property of classical Banach spaces

2000 ◽  
Vol 62 (1) ◽  
pp. 119-134 ◽  
Author(s):  
Aicke Hinrichs
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Real Number ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Average Distance ◽  
Distance Property ◽  
Average Distance Property

A Banach space X has the average distance property (ADP) if there exists a unique real number r such that for each positive integer n and all x1,…,xn in the unit sphere of X there is some x in the unit sphere of X such that .It is known that l2 and l∞ have the ADP, whereas lp fails to have the ADP if 1 ≤ p < 2. We show that lp also fails to have the ADP for 3 ≤ p ≤ ∞. Our method seems to be able to decide also the case 2 < p < 3, but the computational difficulties increase as p comes closer to 2.

scholarly journals The average distance property of classical Banach spaces II

2002 ◽  
Vol 65 (3) ◽  
pp. 511-520 ◽  
Author(s):  
Aicke Hinrichs ◽  
Jörg Wenzel
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Real Number ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Average Distance ◽  
Distance Property ◽  
Average Distance Property

A Banach space X has the average distance property if there exists a unique real number r such that for each positive integer n and all x1,…,xn in the unit sphere of X there is some x in the unit sphere of X such that We show that lp does not have the average distance property if p > 2. This completes the study of the average distance property for lp spaces.

AVERAGE DISTANCE OF SUBSTITUTION NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950097 ◽  
Author(s):  
QIANQIAN YE ◽  
LIFENG XI
Keyword(s):  
Linear Equation ◽  
Metric Spaces ◽  
Deterministic Model ◽  
Geodesic Distance ◽  
Average Distance ◽  
The Self ◽  
Similar Measure ◽  
Self Similar

The substitution network is a deterministic model of evolving self-similar networks. For normalized substitution networks, the limit of metric spaces with respect to networks is a self-similar fractal and the limit of average distances on networks is the integral of geodesic distance of the fractal on the self-similar measure. After some technical handles, we establish the finiteness of integrals and obtain a linear equation set to solve the average distance on the fractal.

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