  The aim of this paper is to shed more light on some recent ideas about
Lyapunov exponents and clarify the formal structures behind these ideas. In
particular, we show that the vector of (averaged) Lyapunov exponents of a
smooth measure-preserving dynamical system can be regarded as the solution of a
vector-valued optimization problem on a space of Riemannian metrics. This
result was first formulated and proved by Jairo Bochi in the language of linear
cocycles and their conjugacies. We go a step further and prove that the
optimization problem is geodesically convex with respect to the $L^2$-metric.
Moreover, we derive some consequences of this fact.
