Abstract
Let X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all bounded continuous real-valued functions in X endowed with the usual sup-norm. We show that the following two properties are equivalent:
-
(a)
X is α-favourable (in the sense of the Banach-Mazur game);
-
(b)
The set of functions h in C(X) for which f + h attains its supremum in X contains a dense and Gδ-subset of the space C(X).
In particular, property (b) has place if X is a compact space or, more generally, if X is homeomorphic to a dense Gδ subset of a compact space.
We show also the equivalence of the following stronger properties:
-
(a′)
X contains some dense completely metrizable subset;
-
(b′)
the set of functions h in C(X) for which f + h has strong maximum in X contains a dense and Gδ-subset of the space C(X).
If X is a complete metric space and f is bounded, then the set of functions h from C(X) for which f + h has both strong maximum and strong minimum in X contains a dense Gδ-subset of C(X).
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First named author would like to acknowledge Robert Deville for inspirational input.
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The first and second author were partially supported by the Bulgarian National Fund for Scientific Research, under grant KP-06-H22/4; The third author was partially supported by the Grant No BG05M2OP001-1.001-0003, financed by the Science and Education for Smart Growth Operational Program (2014-2020) and co-financed by the European Union through the European structural and Investment funds.
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Ivanov, M., Kenderov, P. & Revalski, J. Variational Principles for Maximization Problems with Lower-semicontinuous Goal Functions. Set-Valued Var. Anal 30, 559–571 (2022). https://doi.org/10.1007/s11228-021-00604-1
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DOI: https://doi.org/10.1007/s11228-021-00604-1