Abstract

A subset of Rn\mathbb{R}^n is said to be evenly convex if it is the intersection of some family (possibly empty) of open halfspaces. This class of convex sets was introduced by Fenchel in 1952 in order to extend the polarity theory to nonclosed convex sets. This paper deals with functions with evenly convex epigraphs, the so-called evenly convex functions. We study the main properties of this class of convex functions that contains the important class of lower semicontinuous convex functions. In particular, a characterization of even convexity in terms of lower semicontinuity is given. We also show that the class of evenly convex functions is closed under the main operations.

Author information

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

Margarita M. L. Rodríguez

Dept. of Statistics and Operations Research, University of Alicante, 03080 Alicante, Spain

marga.rodriguez@ua.es

José Vicente-Pérez

Dept. of Statistics and Operations Research, University of Alicante, 03080 Alicante, Spain

jose.vicente@ua.es

Keywords

  • Evenly convex set
  • evenly convex function
  • lsc convex function

Mathematics Subject Classification

26B25, 52A41, 90C25

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Suggested citation

M. M. L. Rodríguez, J. Vicente-Pérez. “On Evenly Convex Functions.” Journal of Convex Analysis 18 (2011), No. 3, 721–736. https://doi.org/10.68381/jca18046

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.