Geodesic
Weak Parallelogram Laws for Banach Spaces
Canadian mathematical bulletin, Tome 19 (1976) no. 3, pp. 269-275

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It has been shown previously that the L p (μ) spaces for 1 < p ≤ 2 satisfy a weak parallelogram law, and the same methods can be used to show that the L p (μ) spaces for 2 ≤ p <∞ satisfy a related weak parallelogram law. This paper obtains several equivalent characterizations of Banach spaces which satisfy one of these two weak parallelogram laws. One such characterization involves the conditions on the moduli of convexity and smoothness analyzed by Lindenstrauss. 
Bynum, W. L. Weak Parallelogram Laws for Banach Spaces. Canadian mathematical bulletin, Tome 19 (1976) no. 3, pp. 269-275. doi: 10.4153/CMB-1976-042-4
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     author = {Bynum, W. L.},
     title = {Weak {Parallelogram} {Laws} for {Banach} {Spaces}},
     journal = {Canadian mathematical bulletin},
     pages = {269--275},
     year = {1976},
     volume = {19},
     number = {3},
     doi = {10.4153/CMB-1976-042-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1976-042-4/}
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