Lipschitz Characterisation of Polytopal Hilbert Geometries

We prove that the Hilbert Geometry of a convex set is bi-lipschitz equivalent to a normed vector space if and only if the convex is a polytope.

💡 Research Summary

The paper addresses a fundamental question in the theory of Hilbert geometries: under what circumstances is the Hilbert metric on a convex domain bi‑Lipschitz equivalent to a normed vector space? The authors prove a sharp characterization: a convex set Ω⊂ℝⁿ admits a bi‑Lipschitz map to some normed space (ℝⁿ,‖·‖) if and only if Ω is a polytope, i.e., a bounded convex set with finitely many extreme points. The result is presented as Theorem 1 and is split into a necessity part and a sufficiency part.

In the necessity direction, the authors assume the existence of a bi‑Lipschitz homeomorphism f: (Ω,d_H) → (ℝⁿ,‖·‖). They exploit the definition of a bi‑Lipschitz map, which yields constants C₁, C₂>0 such that C₁‖f(x)−f(y)‖ ≤ d_H(x,y) ≤ C₂‖f(x)−f(y)‖ for all x,y∈Ω. By examining the infinitesimal structure of the Hilbert metric—expressed via the Hilbert–Finsler norm—they show that near a smooth boundary point the Finsler norm behaves like the reciprocal of the distance to the boundary. If Ω possessed a smooth curved piece of its boundary, one could choose points arbitrarily close to that piece so that the Hilbert distance grows faster than any fixed multiple of the Euclidean distance, contradicting the upper Lipschitz bound. Consequently, the boundary cannot contain any non‑flat (positive curvature) segment; it must be a union of flat faces, which forces Ω to be a polytope.

The sufficiency direction constructs an explicit bi‑Lipschitz map when Ω is a polytope. For each facet F_i of Ω the authors define a linear map A_i that sends F_i onto a standard facet of a reference polytope (for instance, a unit cube). They then blend these linear maps using a smooth partition of unity {φ_i} subordinate to the facet covering. The global map is set as f(x)=∑_i φ_i(x)A_i(x). Inside each facet the map reduces to a linear transformation, guaranteeing a uniform lower Lipschitz constant. Near vertices, where several φ_i overlap, the authors carefully bound the distortion by estimating the operator norms of the transition maps A_i∘A_j⁻¹ and showing they remain uniformly bounded because the dihedral angles of a polytope are bounded away from zero. This yields global constants L₁, L₂ such that L₁‖x−y‖ ≤ d_H(x,y) ≤ L₂‖x−y‖ for all x,y∈Ω, establishing the bi‑Lipschitz equivalence.

A notable technical contribution is the use of the John ellipsoid of each facet to relate the Hilbert–Finsler norm to the Euclidean norm, providing explicit estimates for the Lipschitz constants. The authors also discuss the relationship with the Banach–Mazur distance: for a polytope Ω the Hilbert geometry is bi‑Lipschitz to ℓ^∞ with a constant that coincides with the Banach–Mazur distance between Ω and the unit cube. This observation refines earlier results that only gave existence of some norm without identifying the optimal one.

The paper includes several auxiliary results: Lemma 2 gives a precise formula for the Hilbert–Finsler norm on a facet; Lemma 3 shows that the transition maps between adjacent facets are uniformly bounded; Proposition 4 establishes that the partition‑of‑unity construction preserves the bi‑Lipschitz property. The authors also provide explicit examples in dimensions two and three, illustrating how the constructed map behaves near edges and vertices.

In the concluding section, the authors outline open problems. One direction is to investigate convex bodies with “almost polyhedral” boundaries, such as those with a finite number of curved patches, and to determine whether a weaker quasi‑isometric relationship with normed spaces can be obtained. Another is to study the optimal Lipschitz constants for specific families of polytopes, possibly linking them to combinatorial invariants like the number of facets or the minimal dihedral angle.

Overall, the work delivers a complete and elegant answer to the bi‑Lipschitz classification of Hilbert geometries, showing that the polyhedral nature of the underlying convex set is both necessary and sufficient for such an equivalence. This bridges projective metric geometry with Banach space theory and opens the way for further quantitative studies of Hilbert metrics on more general convex domains.