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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

When does unique local support ensure convexity?


Author: Donald Francis Young
Journal: Trans. Amer. Math. Soc. 347 (1995), 1323-1329
MSC: Primary 26A51; Secondary 26B25
MathSciNet review: 1257125
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Abstract: A basic theorem of convex analysis states that a real-valued function on an open interval of the real line is convex and differentiable if at each point of its domain there exists a unique supporting line. In this paper we show that the same conclusion can be drawn under the weaker hypothesis that there exists a unique locally supporting line at each point. We also show by counterexample that convexity cannot be concluded under analogous circumstances for $ f:S \to \mathbb{R}$, where $ S \subset {\mathbb{R}^n}$ is open and convex, if $ n > 1$.


References [Enhancements On Off] (What's this?)

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  • [5] Donald Francis Young, Local conditions for convexity and upward concavity, College Math. J. 24 (1993), 224-228.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1257125-3
PII: S 0002-9947(1995)1257125-3
Keywords: Convex function, supporting line, Hausdorff Maximality Theorem
Article copyright: © Copyright 1995 American Mathematical Society