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..