Abstract

The invertibility of an order-reversing transform on the class of proper lower semicontinuous convex functions is completely determined by the behavior of its composition with its putative inverse (and of the reverse composition) on the subclasses of continuous affine functions over the primal and dual spaces. This strengthens a recent result of Artstein-Avidan and Milman, which characterizes order-reversing transforms of convex functions as affine adjustments of the Legendre-Fenchel transform.

Author information

Contact details are reproduced from the original publication and may be historical.

Stephen E. Wright

Dept. of Mathematics and Statistics, Miami University, Oxford, OH 45056, U.S.A.

wrightse@muohio.edu

Keywords

  • Locally convex space
  • Legendre-Fenchel transform
  • convex duality

Mathematics Subject Classification

46N10, 52A41

References

  1. S. Artstein-Avidan, V. Milman: The concept of duality in convex analysis, and the characterization of the Legendre transform, Ann. Math. 169 (2009) 661–674.https://doi.org/10.4007/annals.2009.169.661Google Scholar
  2. A. Brøndsted: Conjugate convex functions in topological vector spaces, Mat.-Fys. Medd., Danske Vid. Selsk. 34 (1964) 1–26.Google Scholar
  3. W. Fenchel: On conjugate convex functions, Can. J. Math. 1 (1949) 73–76.https://doi.org/10.4153/CJM-1949-007-xGoogle Scholar
  4. J.-J. Moreau: Fonctions duales et points proximaux dans un espace hilbertien, C. R. Acad. Sci., Paris 255 (1962) 2897–2899.Google Scholar

Suggested citation

S. E. Wright. “Invertibility of Order-Reversing Transforms on Convex Functions.” Journal of Convex Analysis 17 (2010), No. 1, 103–110. https://doi.org/10.68381/jca17009

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.