On the average distance property of spheres in Banach spaces

1994 ◽  
Vol 62 (4) ◽  
pp. 338-344 ◽  
Author(s):  
Reinhard Wolf
Keyword(s):  
Banach Spaces ◽  
Average Distance ◽  
Distance Property ◽  
Average Distance Property

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.

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.

On the Location of Quiescent Prominences with Respect to Coronal Holes

1979 ◽  
Vol 44 ◽  
pp. 209-213
Author(s):  
B. Rompolt
Keyword(s):  
Coronal Hole ◽  
Average Distance ◽  
Coronal Holes

The aim of this contribution is to turn attention to a peculiarity of location of the filaments (quiescent prominences) with respect to the boundaries of the coronal holes. It is generally known that quiescent prominences are located at some distance from the boundary of coronal holes. My intention was to check whether the average distance between the nearest border of a coronal hole and the prominence is comparable to the average horizontal extension of a helmet structure overlying the prominence. As well as, whether this average distance depends upon the orientation of the long axis of the prominence with respect to the nearest boundary of the coronal hole.

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