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On the self-length of two-dimensional Banach spaces

Published online by Cambridge University Press:  17 April 2009

B. Chalmers ,
C. Franchetti  and
M. Giaquinta
B. Chalmers
Affiliation:
Department of Mathematics, University of California, Riverside CA 92521, United States of America
C. Franchetti
Affiliation:
Department of Mathematics, University of California, Riverside CA 92521, United States of America
M. Giaquinta
Affiliation:
Departmento di Matematica Applicata, Università di Firenze, 50139 Firenze, Italy
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The aim of this paper is to prove the following result: if X is a 2-dimensional symmetric real Banach space, then its self-length is greater than or equal to 2π. Moreover, the minimum value 2π is uniquely attained (up to isometries) by euclidean space.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 53 , Issue 1 , February 1996 , pp. 101 - 107
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Chalmers, B.L. and Metcalf, F.T., ‘The determination of minimal projections and extensions in L 1’, Trans. Amer. Math. Soc. 329 (1992), 289–305.Google Scholar
[2]Chalmers, B.L. and Metcalf, F.T., ‘A simple formula showing L 1 is a maximal overspace for two-dimensional real spaces’, Ann. Polon. Math. 56 (1992), 303–309.CrossRefGoogle Scholar
[3]Franchetti, C. and Cheney, E.W., ‘Minimal projections in L 1-space’, Duke Math. J. 43 (1976), 501–510.CrossRefGoogle Scholar
[4]Franchetti, C. and Votruba, G.F., ‘Perimeter, Macphail number and projection constant in Minkowski planes’, Boll. Un. Mat. Ital. B 13 (1976), 560–573.Google Scholar
[5]Garling, D.J.H. and Gordon, Y., ‘Relations between some constants associated with finite dimensional Banach spaces’, Israel J. Math. 9 (1971), 346–361.CrossRefGoogle Scholar
[6]Grünbaum, B., ‘Projection constants’, Trans. Amer. Math. Soc. 95 (1960), 451–465.CrossRefGoogle Scholar
[7]Koenig, H. and Tomczak-Jaegermann, N., ‘Norms of minimal projections’, J. Funct. Anal. 119 (1994), 253–280.CrossRefGoogle Scholar
[8]Lindenstrauss, J., ‘On the extension of operators with a finite-dimensional range’, Illinois J. Math. 8 (1964), 488–499.CrossRefGoogle Scholar
[9]Positselskii, E.D., ‘Projection constants of symmetric spaces’, Math. Notes 15 (1974), 430–435 (Translated from Mat., Zametki 14 (1974), 719–727).CrossRefGoogle Scholar
[10]Yost, D., ‘L 1 contains every two-dimensional normed space’, Ann. Polon. Math. 49 (1988), 17–19.CrossRefGoogle Scholar