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

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.

💡 Research Summary

The paper revisits the Hilbert metric, a classical Finsler‑type distance canonically associated with any open convex domain Ω in Euclidean or projective space. Traditional proofs of its Finsler nature rely on a C² smoothness assumption for ∂Ω and on the Busemann–Mayer theorem, which extracts a norm on each tangent space from the distance function. The authors propose a completely different framework that completely removes any differentiability hypothesis.

The key ingredient is the Funk metric, a naturally defined weak (generally non‑symmetric) distance on Ω. For two points x, y∈Ω, the line through them meets the boundary at points a (closer to x) and b (closer to y); the Funk distance is d_F(x,y)=log(|xb|/|xa|). This definition is purely projective, requires no smoothness, and directly yields a weak Finsler structure: at each point p the unit ball B_F(p) of the associated norm coincides with the whole domain Ω itself. In other words, the tangent space inherits a “tautological” norm that simply identifies vectors with points of Ω.

From this asymmetric structure the authors construct a reversible one by means of harmonic symmetrization. Given a non‑symmetric norm ‖·‖ and its reverse ‖·‖, the harmonic symmetrization is defined by
 ‖v‖_sym = 2 / (1/‖v‖ + 1/‖v‖).
Geometrically this takes the harmonic mean of the two supporting half‑balls and produces a centrally symmetric convex body. Applying this operation pointwise to the Funk unit balls yields new symmetric balls B_H(p). The authors prove that B_H(p) is exactly the unit ball of the Hilbert metric at p, and consequently the Hilbert distance can be written as the symmetrization of the Funk distance:
 d_H(x,y) = ½ ( d_F(x,y) + d_F(y,x) ).

Because the construction never invokes derivatives, all standard Hilbert‑metric properties (triangle inequality, projective invariance, completeness, geodesic convexity, etc.) follow immediately from the corresponding elementary properties of the Funk metric and of harmonic symmetrization. In particular, the Hilbert metric inherits the projective invariance of the Funk metric, while the symmetrization guarantees reversibility.

The paper further explores several consequences. First, it provides explicit formulas for the Funk and Hilbert metrics on convex polygons, showing how the unit balls change from the original polygon (Funk) to its harmonic symmetrization (Hilbert). Second, it extends the theory to higher‑dimensional projective spaces, where the same construction works for any open convex set, regardless of boundary regularity. Third, the authors discuss how harmonic symmetrization clarifies the classical “cross‑ratio” interpretation of the Hilbert distance: the symmetrized cross‑ratio is precisely the harmonic mean of the forward and backward ratios.

Overall, the work establishes a clean, purely metric‑theoretic pathway from a weak Finsler structure (Funk) to the full Hilbert geometry via harmonic symmetrization. This eliminates the need for smoothness assumptions, broadens the class of admissible convex domains, and offers a unifying perspective that may be useful in related areas such as convex analysis, projective geometry, and the study of non‑reversible Finsler spaces.