Adv. Math. Econ. 14, 155-160 (2010)

Convexity of the Lower Partition Range of a Concave Vector Measure^{3}

Nobusumi Sagara^{4}^{1} and Milan Vlach ^{2}

1 Faculty of Economics, Hosei University, 4342, Aihara, Machida, Tokyo 194– 0298, Japan (nsagara@hosei.ac.jp)

2 School of Mathematics and Physics, Charles University, Malostranské náměstí 25, 118 00 Praha 1, Czech Republic (milan.vlach@mfff.cuni.cz)

Received: June 5, 2009 Revised: September 27, 2009

JEL classification: D63

Mathematical Subject Classification (2000): 28B05, 28E10, 91B32

Abstract. This paper investigates a class of nonadditive measures on σ-alge-bras, 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, (1961), “How to cut a cake fairly”, American Mathematical Monthly, vol. 68, pp. 1–17.

Key words: Nonatomic finite measure, Concave measure, Convexity, Lower partition range, Fair division

3The 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 and Technology, Japan.

4Corresponding author.