Geodesic
Asymptotic Smoothness, Convex Envelopes and Polynomial Norms
Journal of convex analysis, Tome 24 (2017) no. 4, pp. 1281-1294
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We introduce a suitable notion of asymptotic smoothness on infinite dimensional Banach spaces, and we prove that, under some structural restrictions on the space, the convex envelope of an asymptotically smooth function is asymptotically smooth. Furthermore, we study convexity and smoothness properties of polynomial norms, and we obtain that a polynomial norm of degree N has modulus of convexity of power type N. 
@article{JCA_2017_24_4_JCA_2017_24_4_a10,
     author = {R. Gonzalo and J. A. Jaramillo and D. Y\'a\~nez},
     title = {Asymptotic {Smoothness,} {Convex} {Envelopes} and {Polynomial} {Norms}},
     journal = {Journal of convex analysis},
     pages = {1281--1294},
     year = {2017},
     volume = {24},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2017_24_4_JCA_2017_24_4_a10/}
}
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R. Gonzalo; J. A. Jaramillo; D. Yáñez. Asymptotic Smoothness, Convex Envelopes and Polynomial Norms. Journal of convex analysis, Tome 24 (2017) no. 4, pp. 1281-1294. http://geodesic.mathdoc.fr/item/JCA_2017_24_4_JCA_2017_24_4_a10/