Whether to keep products segregated (e.g., unbundled) or integrate some or all of them (e.g., bundle) has been a problem of profound interest in areas such as portfolio theory in finance, risk capital allocations in insurance, and marketing of consumer products. Such decisions are inherently...

Whether to keep products segregated (e.g., unbundled) or integrate some or all of them (e.g., bundle) has been a problem of profound interest in areas such as portfolio theory in finance, risk capital allocations in insurance, and marketing of consumer products. Such decisions are inherently...

We propose variants of non-asymptotic dual transcriptions for the functional inequality of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$ f + g + k\circ H \ge h$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo>+</mo> <mi>k</mi> <mo>∘</mo> <mi>H</mi> <mo>≥</mo> <mi>h</mi> </mrow> </math> </EquationSource> </InlineEquation>. The main tool we used consists in purely algebraic formulas on the epigraph of the Legendre-Fenchel transform of the function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$ f + g +...</equationsource></inlineequation></equationsource></equationsource></inlineequation>