Combinatorics Seminar
Raphael Loewy
Technion
Maximal exponents of polyhedral cones
Abstract: Abstract: Let K be a proper (i.e., closed, pointed, full and convex) cone in
$R^n$. We consider an n by n matrix $A$ which is K-primitive. That is,
there exists a positive integer $l$ such that $A^l x$ is in interior of
$K$ for every nonzero $x$ in $K$. The smallest such $l$ is called the
exponent of A, 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 $\gamma(K)$. Our main result is that
for any positive integers $m, n$ such that $3 \le n \le m$, the maximum
value of $\gamma(K)$, as K runs through all n-dimensional polyhedral cones
with m extreme rays, equals
$(n - 1)(m - 1) + 0.5(1+(-1)^{(n-1)m})$.
We will consider various uniqueness issues related to the main result as
well as its connections to known results.
This talk is based on a joint work with Micha Perles and Bit-Shun Tam.
Wednesday February 8, 2012 at 3:00 PM in SEO 512