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Homogeneity of isosceles orthogonality and related inequalities. (English) Zbl 1275.46007

Summary: We study the homogeneity of isosceles orthogonality, which is one of the most important orthogonality types in normed linear spaces, from two viewpoints. On the one hand, we study the relation between homogeneous directions of isosceles orthogonality and other notions including isometric reflection vectors and \(L_2\)-summand vectors and show that a Banach space \(X \) is a Hilbert space if and only if the relative interior of the set of homogeneous directions of isosceles orthogonality in the unit sphere of \(X \) is not empty. On the other hand, we introduce a geometric constant \(NH_X \) to measure the non-homogeneity of isosceles orthogonality. It is proved that \(0 \leq NH_X = 2\), \(NH_X = 0\) if and only if \(X \) is a Hilbert space, and \(NH_X = 2\) if and only if \(X \) is not uniformly non-square.

MSC:

46B20 Geometry and structure of normed linear spaces
46C15 Characterizations of Hilbert spaces
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References:

[1] doi:10.1215/S0012-7094-45-01223-3 · Zbl 0060.26202
[2] doi:10.1215/S0012-7094-35-00115-6 · Zbl 0012.30604
[3] doi:10.1090/S0002-9947-1947-0021241-4
[4] doi:10.1216/rmjm/1022008976 · Zbl 1001.46013
[5] doi:10.1007/s00025-010-0069-6 · Zbl 1219.46020
[6] doi:10.1090/S0002-9939-06-08243-8 · Zbl 1094.46007
[7] doi:10.1017/S1446788700035230
[8] doi:10.1016/j.jmaa.2006.07.046 · Zbl 1154.46302
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