Geodesic
On uniformly smooth renormings of uniformly convex Banach spaces
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIII, Tome 135 (1984), pp. 120-134

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The paper deals with a quantitative aspect of the well-known Enflo-Pisier theorem on the existence of uniformly smooth renormings of superreflexive (in particular, uniformly convex and uniformly non-square) Banach spaces. A typical result: Let the modulus of continuity of a Banach space $X$ with a local unconditional structure satisfy the inequality $\delta_X(\varepsilon)\geqslant c\cdot\varepsilon P$. Then $X$ admits an equivalent $q$-smooth renorming for any $q$ satisfying $$ q\log2/\log[2(1-c\cdot2^{-p/2})]. $$ 
@article{ZNSL_1984_135_a11,
     author = {S. A. Rakov},
     title = {On uniformly smooth renormings of uniformly convex {Banach} spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {120--134},
     publisher = {mathdoc},
     volume = {135},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_135_a11/}
}
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S. A. Rakov. On uniformly smooth renormings of uniformly convex Banach spaces. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIII, Tome 135 (1984), pp. 120-134. http://geodesic.mathdoc.fr/item/ZNSL_1984_135_a11/