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We introduce the notion of strongly <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$t$$</EquationSource> </InlineEquation>-convex set-valued maps and present some properties of it. In particular, a Bernstein–Doetsch and Sierpiński-type theorems for strongly midconvex set-valued maps, as well as a Kuhn-type result are obtained. A representation of strongly <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$t$$</EquationSource> </InlineEquation>-convex...</equationsource></inlineequation></equationsource></inlineequation>
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