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Construction of functions with uniform sublevel sets

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Abstract

In this paper, we study extended real-valued functions with uniform sublevel sets. The sublevel sets are defined by a linear shift of a set in a specified direction. We prove that the class of these functions coincides with the class of Gerstewitz functionals. In this way, we obtain a formula for the construction of such functions. The sublevel sets of Gerstewitz functionals are characterized and illustrated by examples. The results contain statements for translative functions, which are just the functions with uniform sublevel sets considered. The investigated functions are defined on an arbitrary real vector space without assuming any topology or convexity.

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Authors and Affiliations

  1. Faculty of Natural Sciences and Technology, HAWK Hochschule für angewandte Wissenschaft und Kunst Hildesheim/Holzminden/Göttingen, University of Applied Sciences and Arts, Göttingen, Germany

    Petra Weidner

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  1. Petra Weidner
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Correspondence to Petra Weidner.

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Weidner, P. Construction of functions with uniform sublevel sets. Optim Lett 12, 35–41 (2018). https://doi.org/10.1007/s11590-017-1167-0

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  • DOI: https://doi.org/10.1007/s11590-017-1167-0

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