Hilbert Polygonal Metric

• The Hilbert polygonal metric is a projective Finsler metric defined on the interior of convex polygons using cross-ratio formulas, extending the Cayley-Klein hyperbolic model.
• It exhibits properties such as projective invariance, well-defined geodesics, and convex Hilbert metric balls that support efficient computation of Voronoi diagrams and Delaunay triangulations.
• Its applications span convex geometry, computational geometry, machine learning, and optimization, making it a robust tool in both theoretical and practical scenarios.

The Hilbert polygonal metric is a projective, Finsler metric defined on the interior of a convex polygon (or higher-dimensional polytope) that extends the classic Cayley-Klein model of hyperbolic geometry to arbitrary convex domains. It is of central importance in convex geometry, geometric tomography, computational geometry, and has applications in machine learning and optimization. The metric is characterized by its reliance on cross-ratio formulas, strict projective invariance, and the rich polyhedral structure it induces for metric balls, Voronoi diagrams, and Delaunay triangulations.

1. Definition and Basic Properties

Let $\Omega \subset \mathbb{R}^2$ be a convex polygon with interior $\Omega^\circ$. For $p, q \in \Omega^\circ$, $p \neq q$, the Hilbert distance $d_H(p, q)$ is defined using cross-ratios: draw the straight line through $p, q$, meeting the boundary $\partial\Omega$ in two points $a, b$ so that the order along the line is $a, p, q, b$. Then

$$d_H(p, q) = \frac{1}{2}\log\left(\frac{\|q - a\|\cdot\|p - b\|}{\|p - a\|\cdot\|q - b\|}\right),$$

where $\|\cdot\|$ denotes the Euclidean length. The formula is symmetric ($d_H(p,q)=d_H(q,p)$), strictly positive for $p \neq q$, and satisfies the triangle inequality, with equality if and only if the three boundary intersection points corresponding to $p, q, r$ are aligned on a common supporting line (Papadopoulos et al., 2014, Banerjee et al., 3 Mar 2025, Gezalyan et al., 2021).

The metric generalizes to the interior of a full-dimensional convex polytope in $\mathbb{R}^d$ with $m$ facets, each defined by linear equations $L_i = \{x: w_i\cdot x + c_i = 0\}$, via the same cross-ratio construction (Acharya et al., 19 Jan 2026).

Key properties:

• Projective Invariance: $d_H$ is unchanged under any projective mapping sending $\Omega$ to another convex body.
• Geodesics: Every straight segment $[p, q] \subset \Omega^\circ$ is a (possibly unique) geodesic for $d_H$ (Papadopoulos et al., 2014, Gezalyan et al., 2023, Pan, 2017).
• Metric Completeness: The metric space $(\Omega^\circ, d_H)$ is proper; $d_H$ diverges as either argument approaches $\partial\Omega$.

2. Finsler Structure and Infinitesimal Geometry

At each $p \in \Omega^\circ$, the Hilbert metric equips $T_p\Omega \cong \mathbb{R}^2$ with a (generally non-Euclidean) Minkowski norm: $$F_H(p; v) = \frac{1}{2}\left(\frac{1}{t_-} + \frac{1}{t_+}\right)\|v\|_{\text{Eucl}},$$ where $t_+$ is the minimal positive scalar with $p + t_+ v \in \partial \Omega$, and $t_-$ is the minimal positive scalar such that $p - t_- v \in \partial \Omega$. In polygons, these scalars are computed by solving (for each facet)

$$\ell_i(p + \tau v) = c_i \implies \tau = \frac{c_i - \ell_i(p)}{\ell_i(v)}$$

and then taking the minimal positive solutions for each direction (Papadopoulos et al., 2014, Bumpus et al., 2023). The Hilbert norm at $p$ defines a Finsler structure that is strictly convex if and only if $\Omega$ is strictly convex. At points where the tangent directions align with flat faces, the unit sphere is a centrally symmetric polygon with at most $2m$ vertices.

3. Metric Balls and Their Structure

Hilbert metric balls $B(p, r) = \{q \in \Omega^\circ : d_H(p, q) < r\}$ are convex polygons themselves (Papadopoulos et al., 2014, Parepally et al., 2024, Banerjee et al., 3 Mar 2025). In a polygon with $m$ vertices, each ball is a convex polygon with at most $2m$ sides (Gezalyan et al., 2023, Gezalyan et al., 2021). For each vertex $v$ of $\Omega$, a “spoke” through $p$ and $v$ defines two points on the spoke at Hilbert distance $\pm r$ from $p$; the convex hull of these $2m$ points forms $B(p, r)$.

Notably, for regular polygons, these balls are smaller projectively similar polygons. As $p$ approaches the boundary, or as $r \to \infty$, $B(p, r)$ pinches to the boundary, degenerating to polygons with vertices on $\partial \Omega$. For implementation, algorithms compute boundary–spoke intersections and solve the cross-ratio equation to locate the precise radius point along each spoke (Parepally et al., 2024).

4. Geodesics, Uniqueness, and Convexity Properties

Every straight segment in the polygon interior is a geodesic, because the Hilbert metric is preserved under projective transformations and the cross-ratios split additively along such segments (Papadopoulos et al., 2014, Gezalyan et al., 2023, Gezalyan et al., 2021). In strictly convex polygons (no two consecutive sides collinear), geodesics are unique. In the non-strictly convex case (e.g., passing through vertices), geodesics can be concatenations of straight segments meeting at the vertex.

Convexity is a central attribute: all Hilbert balls are convex in the affine sense, since they are defined by intersections of half-planes or angular sectors. The metric's triangle inequality is sharp only when boundary hits for three points are aligned on a supporting line (Papadopoulos et al., 2014, Bumpus et al., 2023).

5. Algorithmic Computation and Visualization

Evaluating $d_H(p, q)$ reduces to finding the boundary intersections $a, b$ of the line through $p, q$. For a polygon with $m$ sides, the computation can be performed in $O(m)$ time through linear scanning, or $O(\log m)$ time with suitable pre-processing (e.g., angle-sorted supporting lines or binary search) (Banerjee et al., 3 Mar 2025, Gezalyan et al., 2021, Gezalyan et al., 2023). Implementations use standard line/segment intersection algorithms, with attention to numerical robustness for nearly parallel or collinear cases. For higher-dimensional polytopes, analogous techniques apply, relying on the facet structure (Acharya et al., 19 Jan 2026).

Visualization software (e.g., browser-based or Ipelets) supports constructing and rendering Hilbert balls, interactive motion (by projective translations), and dynamic Voronoi and Delaunay diagrams (Banerjee et al., 3 Mar 2025, Parepally et al., 2024, Bumpus et al., 2023).

6. Voronoi and Delaunay Structures in the Hilbert Metric

A Voronoi diagram in the Hilbert metric (for sites $P \subset \text{int}\,\Omega$) assigns to each site $p$ the region where $d_H(x, p)$ is minimal among all sites. The Hilbert bisector between two sites is a piecewise conic curve, typically composed of $O(m)$ arc segments per site pair, so the overall Voronoi diagram has $\Theta(mn)$ combinatorial complexity for $n$ sites (Gezalyan et al., 2021, Gezalyan et al., 2023, Bumpus et al., 2023).

The corresponding Delaunay triangulation is the (projective) dual: sites $p, q$ are connected if their Voronoi regions share an arc. This triangulation may not fill the convex hull of $P$, leaving "holes" near $\partial\Omega$ that are filled by "teeth"—triangles involving sites and a boundary vertex. Algorithms for Delaunay construction adapt the Euclidean randomized incremental paradigm with in-circle tests and polygonal ball checks in $O(\log^3 m)$ per in-circle test, with an overall expected running time $O(n(\log n + \log^3 m))$ (Gezalyan et al., 2023).

7. Applications and Higher-Dimensional Generalizations

The Hilbert polygonal metric underlies classification, clustering, and optimization algorithms in machine learning, notably for high-dimensional convex polytopes (Acharya et al., 19 Jan 2026). A large-margin SVM can be formulated in this metric, with balls as convex polytopes and efficient LP-based testing for separability and soft-margin classification. Nearest neighbor and center-based classifiers are supported through explicit isometric embeddings into normed spaces with polyhedral unit balls.

The metric also appears in the study of moduli spaces; for example, the Teichmüller space of punctured surfaces can be parametrized as the interior of a polytope and endowed with the Hilbert metric, yielding geodesic structures and asymptotic growth properties for earthquake flows (Pan, 2017).

• (Papadopoulos et al., 2014) "From Funk to Hilbert Geometry"
• (Gezalyan et al., 2021) "Voronoi Diagrams in the Hilbert Metric"
• (Gezalyan et al., 2023) "Delaunay Triangulations in the Hilbert Metric"
• (Acharya et al., 19 Jan 2026) "Classifiers in High Dimensional Hilbert Metrics"
• (Banerjee et al., 3 Mar 2025) "Software for the Thompson and Funk Polygonal Geometry"
• (Parepally et al., 2024) "Ipelets for the Convex Polygonal Geometry"
• (Bumpus et al., 2023) "Analysis of Dynamic Voronoi Diagrams in the Hilbert Metric"
• (Pan, 2017) "The Hilbert Metric on Teichmüller Space and Earthquake"