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On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces

Mikio Kato;  Lech Maligranda;  Yasuji Takahashi — 2001

Studia Mathematica

Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant C N J ( X ) , and the normal structure coefficient N(X) of Banach spaces X are investigated. Relations between J(X) and J(X*) are given as an answer to a problem of Gao and Lau [16]. Connections between C N J ( X ) and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by the C N J ( X ) -constant, which implies that a Banach space with C N J ( X ) -constant less than 5/4 has the fixed point property.

Clarkson type inequalities and their relations to the concepts of type and cotype.

Mikio Kato;  Lars-Erik. Persson;  Yasuji Takahashi — 2000

Collectanea Mathematica

We prove some multi-dimensional Clarkson type inequalities for Banach spaces. The exact relations between such inequalities and the concepts of type and cotype are shown, which gives a conclusion in an extended setting to M. Milman's observation on Clarkson's inequalities and type. A similar investigation conceming the close connection between random Clarkson inequality and the corresponding concepts of type and cotype is also included. The obtained results complement, unify and generalize several...

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