Abstract
In this paper we introduce the notation of shadowing sets which is a generalization of the notion of separating sets to the family of more than two sets. We prove that \({\bigcap_{i\in I}A_{i}}\) is a shadowing set of the family \({\{A_{i}\}_{i\in I}}\) if and only if \({\sum_{i\in I}A_{i}=\bigvee_{i\in I}\sum_{k\in I\setminus \{i\}}A_{i} + \bigcap_{i\in I}A_{i}}\). It generalizes the theorem stating that \({A\cap B}\) is separating set for A and B if and only if \({A+B=A\cap B+A\vee B}\). In terms of shadowing sets, we give a criterion for an arbitrary upper exhauster to be an exhauster of sublinear function and a criterion for the minimality of finite upper exhausters. Finally we give an example of two different minimal upper exhausters of the same function, which answers a question posed by Vera Roshchina (J Convex Anal, to appear).
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Dedicated to the 75th Birthday of Professor Franco Gianessi.
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Grzybowski, J., Pallaschke, D. & Urbański, R. Reduction of finite exhausters. J Glob Optim 46, 589–601 (2010). https://doi.org/10.1007/s10898-009-9444-9
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DOI: https://doi.org/10.1007/s10898-009-9444-9