The Constants to Measure the Differences Between Birkhoff and Isosceles Orthogonalities

Hiroyasu Mizuguchi


The notion of orthogonality for vectors in inner product spaces is simple, interesting and fruitful.
 When moving to normed spaces, we have many possibilities to extend this notion.
 We consider Birkhoff orthogonality and isosceles orthogonality,
 which are the most used notions of orthogonality.
 In 2006, Ji and Wu introduced a geometric constant $D(X)$
 to give a quantitative characterization of the difference between these two orthogonality types.
 However, this constant was considered only in the unit sphere $S_X$ of the normed space $X$.
 In this paper, we introduce a new geometric constant $IB(X)$ to measure
 the difference between Birkhoff and isosceles orthogonalities in the entire normed space $X$.
 To consider the difference between these orthogonalities, we also treat constant $BI(X)$.

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