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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the stability of almost convex functions


Authors: J. C. Parnami and H. L. Vasudeva
Journal: Proc. Amer. Math. Soc. 97 (1986), 67-70
MSC: Primary 39C05; Secondary 26A51
DOI: https://doi.org/10.1090/S0002-9939-1986-0831389-X
MathSciNet review: 831389
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Abstract: Let $ {\mathbf{R}}$ denote the set of real numbers and $ I$ an open interval of $ {\mathbf{R}}$. A function $ f:I \to R$ is said to be almost $ \delta $-convex iff $ f(tx + (1 - t)y) \leq tf(x) + (1 - t)f(y) + \delta $ holds for all $ (x,y) \in I \times I\backslash N$, where $ N \subset I \times I$ is of measure zero, each $ t \in [0,1]$ and some $ \delta \geq 0$. It is proved that such a function is uniformly close to a convex function almost everywhere.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0831389-X
Article copyright: © Copyright 1986 American Mathematical Society