Collective choice problems on sets in <InlineEquation ID="Equ1"> <EquationSource Format="TEX"> ${\frak R}_+^n$</EquationSource> </InlineEquation> arise naturally in economics. Such problems have been extensively studied both in the theory of revealed preferences (Peters and Wakker, 1991) and in axiomatic bargaining theory under the assumption of convexity. However, our knowledge of...</equationsource></inlineequation>