Extension of Lyapunov’s convexity theorem to subranges
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- by Peng Dai and Eugene A. Feinberg PDF
- Proc. Amer. Math. Soc. 142 (2014), 361-367 Request permission
Abstract:
Consider a measurable space with a finite vector measure. This measure defines a mapping of the $\sigma$-field into a Euclidean space. According to Lyapunov’s convexity theorem, the range of this mapping is compact and, if the measure is atomless, this range is convex. Similar ranges are also defined for measurable subsets of the space. We show that the union of the ranges of all subsets having the same given vector measure is also compact and, if the measure is atomless, it is convex. We further provide a geometrically constructed convex compactum in the Euclidean space that contains this union. The equality of these two sets, which holds for two-dimensional measures, can be violated in higher dimensions.References
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Additional Information
- Peng Dai
- Affiliation: Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, New York 11794-3600
- Email: Peng.Dai@outlook.com
- Eugene A. Feinberg
- Affiliation: Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, New York 11794-3600
- Email: Eugene.Feinberg@stonybrook.edu
- Received by editor(s): February 21, 2011
- Received by editor(s) in revised form: February 28, 2012
- Published electronically: September 20, 2013
- Additional Notes: This research was partially supported by NSF grants CMMI-0900206 and CMMI-0928490.
- Communicated by: Mark M. Meerschaert
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 361-367
- MSC (2010): Primary 60A10, 28A10
- DOI: https://doi.org/10.1090/S0002-9939-2013-11864-2
- MathSciNet review: 3119209