Geodesic
A universal modulus for normed spaces
Studia Mathematica, Tome 127 (1998) no. 1, pp. 21-46 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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We define a handy new modulus for normed spaces. More precisely, given any normed space X, we define in a canonical way a function ξ:[0,1)→ ℝ which depends only on the two-dimensional subspaces of X. We show that this function is strictly increasing and convex, and that its behaviour is intimately connected with the geometry of X. In particular, ξ tells us whether or not X is uniformly smooth, uniformly convex, uniformly non-square or an inner product space. 
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Carlos Benítez; Krzysztof Przesławski; David Yost. A universal modulus for normed spaces. Studia Mathematica, Tome 127 (1998) no. 1, pp. 21-46. doi: 10.4064/sm-127-1-21-46

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