Taiwanese Journal of Mathematics
- Taiwanese J. Math.
- Volume 17, Number 6 (2013), 1967-1982.
NO DICE THEOREM ON SYMMETRIC CONES
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  gave an affirmative answer to the conjecture in the space of positive definite matrices. In this paper, inspired by the work of Holbrook  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.
Taiwanese J. Math., Volume 17, Number 6 (2013), 1967-1982.
First available in Project Euclid: 10 July 2017
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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]
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