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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Maximal exponents of polyhedral cones (III)


Authors: Raphael Loewy, Micha A. Perles and Bit-Shun Tam
Journal: Trans. Amer. Math. Soc. 365 (2013), 3535-3573
MSC (2010): Primary 15B48, 47A65, 05C50, 52B99
Published electronically: March 18, 2013
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Abstract: Let $ K$ be a proper (i.e., closed, pointed, full, convex) cone in $ \mathbb{R}^n$. An $ n\times n$ matrix $ A$ is said to be $ K$-primitive if $ AK\subseteq K$ and there exists a positive integer $ k$ such that $ A^k(K \setminus \{ 0 \}) \subseteq $ int$ K$; the least such $ k$ is referred to as the exponent of $ A$ and is denoted by $ \gamma (A)$. For a polyhedral cone $ K$, the maximum value of $ \gamma (A)$, taken over all $ K$-primitive matrices $ A$, is denoted by $ \gamma (K)$. It is proved that for any positive integers $ m,n, 3 \le n \le m$, the maximum value of $ \gamma (K)$, as $ K$ runs through all $ n$-dimensional polyhedral cones with $ m$ extreme rays, equals $ (n-1)(m-1)+\frac {1}{2}\left (1+(-1)^{(n-1)m}\right )$. For the $ 3$-dimensional case, the cones $ K$ and the corresponding $ K$-primitive matrices $ A$ such that $ \gamma (K)$ and $ \gamma (A)$ attain the maximum value are identified up to respectively linear isomorphism and cone-equivalence modulo positive scalar multiplication.


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Additional Information

Raphael Loewy
Affiliation: Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel
Email: loewy@techunix.technion.ac.il

Micha A. Perles
Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel
Email: perles@math.huji.ac.il

Bit-Shun Tam
Affiliation: Department of Mathematics, Tamkang University, Tamsui, Taiwan 251, Republic of China
Email: bsm01@mail.tku.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05879-5
PII: S 0002-9947(2013)05879-5
Keywords: Cone-preserving map, $K$-primitive matrix, Exponents, polyhedral cone, exp-maximal cone, exp-maximal $K$-primitive matrix, cone-equivalence
Received by editor(s): August 2, 2011
Published electronically: March 18, 2013
Additional Notes: The third author (the corresponding author) was supported by the National Science Council of the Republic of China (Grant No. NSC 98–2115–M–032 –007 –MY3)
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.