For every finite subset F of the unit sphere S of a Banach space and for every x 2 S consider the average distance μ(F, x) between the points of F and x. The set of average numbers for a Banach space X is the closed interval [μ1(X), μ2(X)], where μ1(X) = supF infx μ(F, x) and μ2(X) = infF supx μ(F, x) as F runs over all finite subsets of S and x runs over S. First we prove some general results about the average numbers, including the fact that μ1 and μ2 depend continuously on the Banach-Mazur distance between spaces. Then we compute μ1(X) and μ2(X) and other related parameters for some Banach spaces X such as c0, l1, l1, L[0, 1] and C[0, 1]. We also discuss how the average numbers are related to the modulus of convexity, to orthogonality, and to the existence of extreme points. Finally, we prove that for every Banach space X the average numbers of the bidual X are also average numbers of X.
On the average distance property and the size of the unit sphere.
CASINI, EMANUELE GIUSEPPE;
1998-01-01
Abstract
For every finite subset F of the unit sphere S of a Banach space and for every x 2 S consider the average distance μ(F, x) between the points of F and x. The set of average numbers for a Banach space X is the closed interval [μ1(X), μ2(X)], where μ1(X) = supF infx μ(F, x) and μ2(X) = infF supx μ(F, x) as F runs over all finite subsets of S and x runs over S. First we prove some general results about the average numbers, including the fact that μ1 and μ2 depend continuously on the Banach-Mazur distance between spaces. Then we compute μ1(X) and μ2(X) and other related parameters for some Banach spaces X such as c0, l1, l1, L[0, 1] and C[0, 1]. We also discuss how the average numbers are related to the modulus of convexity, to orthogonality, and to the existence of extreme points. Finally, we prove that for every Banach space X the average numbers of the bidual X are also average numbers of X.
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