A note on the modulus of convexity
Glasgow mathematical journal, Tome 22 (1981) no. 2, pp. 157-158
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In [1, Corollary 5], Figiel gives an elegant demonstration that the modulus ofconvexity δ in real Banach space X is nondecreasing, whereIt is deduced from this that in fact δ(ɛ)/ɛ is nondecreasing [Proposition 3]. During the course of the proof [Lemma 4] it is stated that if v ∊ Sx is a local maximum on Sx of φ ∈Sx*, then v is a global maximum (φ(v) = 1). This is false; it could be that v is a global minimum. It is easy to construct such an example in R2 endowed with the maximum norm. What is true is that v is a global maximum of |φ|.
Plant, Andrew T. A note on the modulus of convexity. Glasgow mathematical journal, Tome 22 (1981) no. 2, pp. 157-158. doi: 10.1017/S0017089500004614
@article{10_1017_S0017089500004614,
author = {Plant, Andrew T.},
title = {A note on the modulus of convexity},
journal = {Glasgow mathematical journal},
pages = {157--158},
year = {1981},
volume = {22},
number = {2},
doi = {10.1017/S0017089500004614},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500004614/}
}
[1] 1.Figiel, T., On the moduli of convexity and smoothness, Studia Math.. 56 (1976), 121–155. Google Scholar | DOI
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