Postprint version. Published in Journal of Mathematical Analysis and Applications, Volume 281, Issue 1, January 1, 2003, pages 346-361. Copyright © 2003 Elsevier. The definitive version is available at http://dx.doi.org/10.1016/S0022-247X(03)00107-0.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
For fair-division or cake-cutting problems with value functions which are normalized positive measures (i.e., the values are probability measures) maximin-share and minimax-envy inequalities are derived for both continuous and discrete measures. The tools used include classical and recent basic convexity results, as well as ad hoc constructions. Examples are given to show that the envy-minimizing criterion is not Pareto optimal, even if the values are mutually absolutely continuous. In the discrete measure case, sufficient conditions are obtained to guarantee the existence of envy-free partitions.