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

Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry