| Co-Investigator(Kenkyū-buntansha) |
SATO Ryotaro OKAYAMA UNIVERSITY, GRADUATE SCHOOL OF NATURAL SCIENCE AND TECHNOLOGY, PROFESSOR, 大学院・自然科学研究科, 教授 (50077913)
KAWABATA Hiroaki FACULTY OF COMPUTER SCIENCE AND SYSTEM ENGINEERING, PROFESSOR, 情報工学部, 教授 (70081271)
TAKAHASHI Hiromitsu FACULTY OF COMPUTER SCIENCE AND SYSTEM ENGINEERING, PROFESSOR, 情報工学部, 教授 (30109889)
KATO Mikio KYUSHU INSTITUTE OF TECHNOLOGY, FACULTY OF ENGINEERING, PROFESSOR, 工学部, 教授 (50090551)
TAKAHASI Sin-ei YAMAGATA UNIVERSITY, FACULTY OF ENGINEERING, PROFESSOR, 工学部, 教授 (50007762)
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| Research Abstract |
In this research we investigated some geometrical properties of Banach spaces X in connection with classical norm inequalities and their generalizations. We also considered some generalizations of James and Schaffer constants, and showed that some geometrical properties of X can be described in terms of these constants. The uniform convexity and non-squareness of ψ-direct sums of Banach spaces were also investigated.
The main results are stated as follows :
1. Norm inequalities and geometry of Banach spaces :
We consider some generalizations of the classical norm inequalities for Banach spaces X , and characterize some geometrical properties of X in terms of these inequalities. In particular, we prove the exact relations between weighted Clarkson type inequalities and the concepts of p-uniform smoothness and q-uniform convexity, and give the optimal 2-uniform convexity inequalities for concrete Banach spaces. Some extensions of Hanner type and Hlawka type inequalities are also given.
2. James, Schaffer type constants and geometry of Banach spaces :
We introduce the James type constant J_<X, t>(τ) and the Schaffer type constant S_<X, t>(τ) for Banach spaces X, and investigate fundamental properties of these constants. We show that some properties of X such as uniform convexity, smoothness and non-squareness can be described in terms of the constants J_<X, t>(τ) and S_<X, t>(τ). Some examples of concrete Banach spaces with the calculation of these constants are also given.
3. Absolute norms and ψ-direct sums of Banach spaces :
We consider the ψ-direct sum X【symmetry】 _ψ Y of Banach spaces X and Y, and show that X【symmetry】_ψY is uniformly convex (locally uniformly convex) if and only if X, Y are uniformly convex (locally uniformly convex) and ψ is strictly convex. Similar characterizations of strict convexity and uniform non-squareness for X 【symmetry】TJ_ψ Y are also given.
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