The notion of almost convexity is studied and an extension of Kuczma's theorem, originally proved in finite-dimensional spaces, is presented. The phrase "almost" is meant in the sense of abstract σ-ideals. The main result also generalizes the theorem proved in Jarczyk and Laczkovich (Math Inequal Appl 13:217-225, 2009).

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Witold Jarczyk obtained his PhD and habilitation from the Silesian University in 1983 and 1993, respectively. Presently, he is professor of mathematics at the University of Zielona Góra in Poland. His interests lie in functional equations and inequalities, iteration theory and discrete dynamical systems. Weinian Zhang received his PhD from the Peking University in 1990. Presently, he is professor of mathematics at the Sichuan University in China. His interests lie in bifurcation theory of differential equations, iteration theory and dynamical systems.

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