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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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When does unique local support ensure convexity?
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by Donald Francis Young PDF
Trans. Amer. Math. Soc. 347 (1995), 1323-1329 Request permission

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
    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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 1323-1329
  • MSC: Primary 26A51; Secondary 26B25
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1257125-3
  • MathSciNet review: 1257125