Abstract

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

Author information

Contact details are reproduced from the original publication and may be historical.

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 Yáñez

Dep. de Matemáticas, Escuela de Ingenierías Industriales, Universidad de Extremadura, 06006 Badajoz, Spain

dyanez@unex.es

Keywords

  • Convex envelope
  • asymptotic smoothness
  • polynomial norm

Suggested citation

R. Gonzalo, J. A. Jaramillo, D. Yáñez. “Asymptotic Smoothness, Convex Envelopes and Polynomial Norms.” Journal of Convex Analysis 24 (2017), No. 4, 1281–1294. https://doi.org/10.68381/jca24073

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press.