Journal of Convex Analysis 24 (2017), No. 4, 1281--1294
Copyright Heldermann Verlag 2017
Asymptotic Smoothness, Convex Envelopes and Polynomial Norms
Raquel Gonzalo
Dep. de Matemática Aplicada, Universidad Politécnica, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain
rngonzalo@fi.upm.es
Jesús A. Jaramillo
Instituto de Matemática Interdiscliplinar, Dep. de Análisis Matemático, Universidad Complutense, 28040 Madrid, Spain
jaramil@mat.ucm.es
Diego Yáńez
Dep. de Matemáticas, Escuela de Ingenierías Industriales, Universidad de Extremadura, 06006 Badajoz, Spain
dyanez@unex.es
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.
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