Asymptotic Smoothness, Convex Envelopes and Polynomial Norms

TL;DR

This paper introduces a new concept of asymptotic smoothness in infinite-dimensional Banach spaces, explores how convex envelopes preserve this property, and analyzes the convexity and smoothness of polynomial norms, revealing their modulus of convexity.

Contribution

It defines asymptotic smoothness in Banach spaces, proves convex envelopes preserve this property under certain conditions, and characterizes polynomial norms' convexity and smoothness properties.

Findings

Convex envelope of asymptotically smooth functions remains asymptotically smooth.

Polynomial norms of degree N have modulus of convexity of power type N.

Structural restrictions are necessary for the preservation of asymptotic smoothness.

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.