Taiwanese Journal of Mathematics

NO DICE THEOREM ON SYMMETRIC CONES

Sangho Kum, Hosoo Lee, and Yongdo Lim

Full-text: Open access

Abstract

The monotonicity of the least squares mean on the Riemannian manifold of positive definite matrices, conjectured by Bhatia and Holbrook and one of key axiomatic properties of matrix geometric means, was recently established based on the Strong Law of Large Number [14, 4]. A natural question concerned with the S.L.L.N is so called the no dice conjecture. It is a problem to make a construction of deterministic sequences converging to the least squares mean without any probabilistic arguments. Very recently, Holbrook [7] gave an affirmative answer to the conjecture in the space of positive definite matrices. In this paper, inspired by the work of Holbrook [7] and the fact that the convex cone of positive definite matrices is a typical example of a symmetric cone (self-dual homogeneous convex cone), we establish the no dice theorem on general symmetric cones.

Article information

Source
Taiwanese J. Math., Volume 17, Number 6 (2013), 1967-1982.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706280

Digital Object Identifier
doi:10.11650/tjm.17.2013.2944

Mathematical Reviews number (MathSciNet)
MR3141869

Zentralblatt MATH identifier
1295.47008

Subjects
Primary: 47A64: Operator means, shorted operators, etc. 17C50: Jordan structures associated with other structures [See also 16W10] 15B48: Positive matrices and their generalizations; cones of matrices 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]

Keywords
least squares mean symmetric cone no dice theorem

Citation

Kum, Sangho; Lee, Hosoo; Lim, Yongdo. NO DICE THEOREM ON SYMMETRIC CONES. Taiwanese J. Math. 17 (2013), no. 6, 1967--1982. doi:10.11650/tjm.17.2013.2944. https://projecteuclid.org/euclid.twjm/1499706280


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