Maximal exponents of polyhedral cones (III)

Let K be a proper (i.e., closed, pointed, full, convex) cone in Rⁿ. An n × n matrix A is said to be K-primitive if AK ⊆ K and there exists a positive integer k such that A^k(K\{0}) ⊆ int K; the least such k is referred to as the exponent of A and is denoted by γ(A). For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is denoted by γ(K). It is proved that for any positive integers m, n, 3 ≤ n ≤ m, the maximum value of γ(K), as K runs through all n-d dimensional polyhedral cones with m extreme rays, equals (n - 1)(m - 1) + ¹². For the 3-dimensional case, the cones K and the corresponding K-primitive matrices A such that γ(K) and γ(A) attain the maximum value are identified up to respectively linear isomorphism and cone-equivalence modulo positive scalar multiplication.