JIPAM

Outer $ \gamma$-Convex Functions on a Normed Space  
 
  Authors: Phan Thanh An,  
  Keywords: Convexity, Epigraph, Jensen inequality, Outer $ gamma$-convex set, Outer $ gamma$-convex function  
  Date Received: 22/03/05  
  Date Accepted: 28/06/05  
  Subject Codes:

26A51, 26B25, 52A41.

  
  Editors: Alexander M. Rubinov (1940-2006),  
 
  Abstract:

For some given positive $ gamma$, a function $ f$ is called outer $ gamma$ -convex if it satisfies the Jensen inequality $ f(z_i)leq(1- lambda_i)f(x_0)+lambda_i f(x_1)$ for some $ z_0colon=x_0,z_1,...,z_k colon=x_1in [x_0,x_1]$ satisfying $ Vert z_i - z_{i+1}Vertlegamma$, where $ lambda_icolon=Vert x_0-z_iVert/Vert x_0-x_1Vert, i=1,2,...,k-1$. Though the Jensen inequality is only required to hold true at some points (although the location of these points is uncertain) on the segment $ [x_0,x_1]$, such a function has many interesting properties similar to those of classical convex functions. Among others it is shown that, if the infimum limit of an outer $ gamma$-convex function attains $ -infty$ at some point then this propagates to other points, and under some assumptions, a function is outer $ gamma$-convex iff its epigraph is an outer $ gamma$-convex set. ;


This article was printed from JIPAM
http://jipam.vu.edu.au

The URL for this article is:
http://jipam.vu.edu.au/article.php?sid=558