Maximal exponents of polyhedral cones
Speaker:\n\nProf. R. Loewy\n\n\nAbstract:\n\nLet K be a proper (i.e., closed, pointed, full and convex) cone in R^n. We consider A\u2208R^(n\xd7n) which is K-primitive, that is, there exists a positive integer l such that A^l.x \u2208 int K for every 0\u2260x\u2208K. The smallest such l is called the exponent of A, denoted by \u03b3(A).\r\rFor a polyhedral cone K, the maximum value of \u03b3(A), taken over all K-primitive matrices A, is denoted by \u03b3(K). Our main result is that for any positive integers m,n, 3 \u2264 n \u2264 m, the maximum value of \u03b3(K), as K runs through all n-dimensional polyhedral cones with m extreme rays, equals\r\r( n - 1 )( m - 1 ) + \xbd( 1 + (-1)^{(n-1)m} ).\r\rWe will consider various uniqueness issues related to the main result as well as its connections to known results.\r\rThis talk is based on a joint work with Micha Perles and Bit-Shun Tam.