Self-concordant barriers for hyperbolic means

The geometric mean and the function (det(.))(1/m) (on the m-by-m positive d efinite matrices) are examples of "hyperbolic means": functions of the form p(1/m) where p is a hyperbolic polynomial of degree m. (A homogeneous poly nomial p is "hyperbolic" with respect to a vector d if the polynomial t --> p(x + td) has only real roots for every vector x.) Any hyperbolic mean is positively homogeneous and concave (on a suitable domain): we present a sel f-concordant harrier for its hypograph. with barrier parameter O(m(2)). Our approach is direct, and shows. for example, that the function -m log(det(. ) - 1) is an m(2)-self-concordant barrier on a natural domain. Such barrier s suggest novel interior point approaches to convex programs involving hype rbolic mean..