Abstract

Given a topological space XX, an interval I⊆RI\subseteq {\bf R} and five continuous functions φ,ψ,ω:X→R\varphi, \psi, \omega: X\to {\bf R}, α,β:I→R\alpha, \beta:I\to {\bf R}, we are interested in the infimum of the function Φ:X→]−∞,+∞]\Phi:X\to ]-\infty,+\infty] defined by Φ(x)=sup⁡λ∈I(α(λ)φ(x)+β(λ)ψ(x))+ω(x) .\Phi(x)=\sup_{\lambda\in I}(\alpha(\lambda)\varphi(x)+\beta(\lambda)\psi(x))+\omega(x)\,. Using a recent minimax theorem of the author [see Minimax theorems in a fully non-convex setting, J. Nonlinear Var. Analysis 3 (2019) 45-52], we build a general scheme which provides the exact value of inf⁡XΦ\inf_X\Phi for a large class of functions Φ\Phi. When additional compactness conditions are satisfied, our scheme provides also the existence of (explicitly detected) functions γ,η:X→R\gamma, \eta:X\to {\bf R} such that, for some x~∈X\tilde x\in X, one has γ(x~)φ(x~)+η(x~)ψ(x~)+ω(x~)=inf⁡x∈X(γ(x~)φ(x)+η(x~)ψ(x)+ω(x)) .\gamma(\tilde x)\varphi(\tilde x)+\eta(\tilde x)\psi(\tilde x)+\omega(\tilde x) = \inf_{x\in X}(\gamma(\tilde x)\varphi(x)+\eta(\tilde x)\psi(x)+\omega(x))\,.

Author information

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

Biagio Ricceri

Department of Mathematics and Informatics, University of Catania, Italy

ricceri@dmi.unict.it

Keywords

  • Infimum
  • minimax
  • inf-connectdness
  • inf-compactness

Mathematics Subject Classification

49J35, 90C47

References

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Suggested citation

B. Ricceri. “On the Infimum of the Upper Envelope of Certain Families of Functions.” Journal of Convex Analysis 32 (2025), No. 3, 789–800. https://doi.org/10.68381/jca32039

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.