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  Volume 6, Issue 3, Article 85
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),  

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.

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