Abstract
This paper investigates a class of nonadditive measures on σ-algebras, named concave measures, to establish a Lyapunov-type convexity theorem. To this end, we introduce convex combinations of measurable sets in terms of a nonatomic vector measure and demonstrate the convexity of the lower partition range of a concave vector measure. The main result is applied to a fair division problem along the lines of L.E. Dubins and E.H. Spanier (Amer Math Monthly 68:1–17, 1961).
Received: June 5, 2009
Revised: September 27, 2009
JEL classification: D63
Mathematics Subject Classification (2000): 28B05, 28E10, 91B32
∗ The authors are grateful to the editor of this journal for his helpful comments. This research is supported by a Grant-in-Aid for Scientific Research (No. 18610003) from the Ministry of Education Culture Sports, Science Technology Japan.
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Sagara, N., Vlach, M. (2010). Convexity of the lower partition range of a concave vector measure. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 13. Springer, Tokyo. https://doi.org/10.1007/978-4-431-99490-9_6
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DOI: https://doi.org/10.1007/978-4-431-99490-9_6
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