  David Hilbert discovered in 1895 an important metric that is canonically
associated to any convex domain $\Omega$ in the Euclidean (or projective)
space. This metric is known to be Finslerian, and the usual proof assumes a
certain degree of smoothness of the boundary of $\Omega$ and refers to a
theorem by Busemann and Mayer that produces the norm of a tangent vector from
the distance function. In this paper, we develop a new approach for the study
of the Hilbert metric where no differentiability is assumed. The approach
exhibits the Hilbert metric on a domain as a symmetrization of a natural weak
metric, known as the Funk metric. The Funk metric is described as a
tautological weak Finsler metric, in which the unit ball at each tangent space
is naturally identified with the domain $\Omega$ itself. The Hilbert metric is
then identified with the reversible tautological weak Finsler structure on
$\Omega$, and the unit ball at each point is described as the harmonic
symmetrization of the unit ball of the Funk metric. Properties of the Hilbert
metric then follow from general properties of harmonic symmetrizations of weak
Finsler structures.
