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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 $n > 1$.

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    N. Bourbaki, Fonctions d’une variable réelle, Hermann, Paris, 1958. A. M. Bruckner, A general convexity criterion, Glas. Mat. Ser. III 13(33) (1978), 231-235. A. Wayne Roberts and Dale E. Varburg, Convex functions, Academic Press, New York, 1973. Clifford E. Weil, Monotonicity, convexity and symmetric derivates, Trans. Amer. Math. Soc. 221 (1976), 225-237. Donald Francis Young, Local conditions for convexity and upward concavity, College Math. J. 24 (1993), 224-228.

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Keywords: Convex function, supporting line, Hausdorff Maximality Theorem
Article copyright: © Copyright 1995 American Mathematical Society