[Paper Review] Hilbert geometry of polytopes

Andreas Bernig|arXiv (Cornell University)|Dec 5, 2008

Point processes and geometric inequalities4 citations

TL;DR

This paper proves that the Hilbert metric on the interior of any convex polytope is bilipschitz equivalent to a normed vector space of the same dimension. The authors construct an explicit diffeomorphism Φ(x) = ∑ log fᵢ(x) grad fᵢ mapping the polytope's interior to the dual space, showing both Lipschitz continuity and a lower bound on the differential's norm, which establishes the bilipschitz equivalence.

ABSTRACT

It is shown that the Hilbert metric on the interior of a convex polytope is bilipschitz to a normed vector space of the same dimension.

Motivation & Objective

• To resolve the open question of whether the Hilbert metric on a convex polytope is bilipschitz equivalent to a normed vector space.
• To establish a quantitative geometric equivalence between Hilbert geometry and finite-dimensional normed spaces for polytopal domains.
• To provide a constructive proof using a specific diffeomorphism based on the defining affine functions of the polytope.
• To extend prior results on Hilbert metrics of polygons and C¹¹ bodies to general polytopes, clarifying the bilipschitz classification.

Proposed method

• Define the Hilbert metric d(x,y) via cross-ratio on the line through x and y intersecting the boundary ∂P.
• Construct the map Φ: int(P) → V* by Φ(x) = ∑ᵢ log fᵢ(x) dfᵢ, where fᵢ are the affine functions defining the polytope.
• Prove Lipschitz continuity of Φ using bounds on the log differences |log fᵢ(x) - log fᵢ(y)| and the gradient norms of fᵢ.
• Establish injectivity of Φ via strict positivity of the inner product ⟨Φ(x) - Φ(y), x - y⟩ when x ≠ y.
• Use induction on face dimensions to prove the differential dΦ satisfies a lower bound ‖dΦ(v)‖₂ ≥ C_F ‖v‖ for v in tangent spaces.
• Apply geometric estimates near faces and use quotient spaces to reduce to lower-dimensional polyhedral cones, ensuring uniform lower bounds on the differential.

Experimental results

Research questions

• RQ1 Is the Hilbert metric on a convex polytope bilipschitz equivalent to a normed vector space?
• RQ2 Can a canonical diffeomorphism be constructed that realizes this bilipschitz equivalence?
• RQ3 What is the behavior of the Hilbert metric near the boundary of a polytope, and how does it relate to the geometry of the defining affine functions?
• RQ4 Does the bilipschitz property hold uniformly across all tangent directions, even near lower-dimensional faces?
• RQ5 How does the differential of the proposed map Φ behave in relation to the Finsler norm and the Euclidean norm on the tangent space?

Key findings

• The Hilbert metric on the interior of any compact convex polytope is bilipschitz equivalent to a normed vector space of the same dimension.
• The map Φ(x) = ∑ᵢ log fᵢ(x) grad fᵢ is a bilipschitz diffeomorphism from int(P) to V*.
• The bilipschitz constant depends only on the number of facets m, the maximum gradient norm R of the defining affine functions, and the diameter D of the polytope.
• Lipschitz continuity of Φ is established via the bound ‖Φ(x) - Φ(y)‖₂ ≤ 2mR d(x,y).
• The differential dΦ satisfies a uniform lower bound ‖dΦ(v)‖₂ ≥ C_F ‖v‖ for all tangent vectors v, with C_F depending on the face dimension and the geometry of P.
• The result holds uniformly across all compact subsets of int(P), with the lower bound on dΦ being uniform near each face of the polytope.