Geometry

Hilbert geometry

A bounded convex region $\Omega$ becomes a metric space once you define $d_\Omega$ by the cross-ratio of where a line exits its boundary — David Hilbert's 1895 construction, which reduces to the classical Klein model of hyperbolic geometry exactly when $\Omega$ is an ellipse, and to a genuinely new non-Riemannian geometry otherwise.

IntuitionDistances that grow as you approach the edge

Take any bounded convex region in the plane — a disk, an ellipse, a square, a hexagon, anything that does not stretch out to infinity and has no dents. Call it $\Omega$ . Pick two points $x$ and $y$ inside $\Omega$ , draw the straight line through them, and let it exit the region at two boundary points $a$ and $b$ , so that the four points sit in the order $a, x, y, b$ along the line. Hilbert's trick is to measure the distance between $x$ and $y$ not by a ruler, but by how the two segments $ay$ and $bx$ compare to the two segments $ax$ and $by$ — a ratio of ratios called a cross-ratio. Concretely, inside the unit disk $\Omega = \{x^2+y^2<1\}$ , take the center $x=(0,0)$ and a point $y=(0.6,0)$ on the horizontal radius. The line through them meets the circle at $a=(-1,0)$ and $b=(1,0)$ . Measuring along that line, $|ay|=1.6$ , $|bx|=1$ , $|ax|=1$ , $|by|=0.4$ , so the cross-ratio is $\frac{1.6\times 1}{1\times 0.4}=4$ , and Hilbert's distance is $\tfrac12\ln 4=\ln 2\approx 0.693$ . Now slide $y$ outward, say to $(0.9,0)$ : $|ay|=1.9$ , $|by|=0.1$ , and the cross-ratio jumps to $\frac{1.9\times1}{1\times0.1}=19$ , giving distance $\tfrac12\ln 19\approx 1.47$ — more than double, even though $y$ only moved a little in ordinary Euclidean terms. The closer a point sits to the boundary of $\Omega$ , the farther away Hilbert's metric pushes it, exactly the behavior that makes the classical Klein model of hyperbolic geometry work: in that model, the boundary circle represents points 'at infinity', infinitely far from the center.

A solid cube: an example of a bounded convex body in space. Note that Hilbert's construction itself is usually pictured in 2D (a flat convex region such as a disk or polygon) — this 3D cube is not literally a Hilbert-geometry picture, but illustrates the general notion of a bounded convex domain that the construction generalizes to in any dimension.

UndergraduateHilbert's construction: distance from a cross-ratio

Definition: The Hilbert metric

Let $\Omega$ be a bounded, open, convex subset of $\mathbb{R}^n$ (equivalently, an affine chart of a properly convex open set in real projective space $\mathbb{RP}^n$ ). For distinct points $x, y \in \Omega$ , let the line through $x$ and $y$ meet the boundary $\partial\Omega$ at two points $a$ and $b$ , labelled so the four points occur in the order $a, x, y, b$ along the line. David Hilbert defined, in a short 1895 note written as a letter to Felix Klein, the Hilbert distance $d_\Omega(x,y) = \frac12 \ln[a,x,y,b] = \frac12 \ln\frac{|ay|\,|bx|}{|ax|\,|by|},$ together with $d_\Omega(x,x)=0$ , where $[a,x,y,b]$ denotes the cross-ratio of the four collinear points and $|\cdot|$ denotes ordinary Euclidean length along that line (any affine parametrization gives the same ratio, since a cross-ratio is unchanged by reparametrizing the line).

d_\Omega(x,y) = \frac12 \ln[a,x,y,b] = \frac12 \ln\frac{|ay|\,|bx|}{|ax|\,|by|}

Theorem: The Hilbert distance is a genuine metric, with straight segments as geodesics

For $\Omega$ as above, $d_\Omega$ is a metric on $\Omega$ : it is symmetric ( $d_\Omega(x,y)=d_\Omega(y,x)$ ), non-negative, and $d_\Omega(x,y)=0$ if and only if $x=y$ . Moreover, for any $x,z\in\Omega$ and any point $y$ on the straight segment $[x,z]$ , $d_\Omega(x,y)+d_\Omega(y,z)=d_\Omega(x,z)$ ; consequently the ordinary Euclidean straight-line segment $[x,z]$ (traversed at the appropriate non-uniform speed) is a geodesic of $(\Omega,d_\Omega)$ , and $d_\Omega$ satisfies the triangle inequality with equality exactly along such segments.

Why is it true?

The formula $d_\Omega(x,y) = \frac12\ln[a,x,y,b]$ is symmetric under swapping the roles of $x$ and $y$ together with $a$ and $b$ , because relabelling turns $\frac{|ay||bx|}{|ax||by|}$ into itself; and because $a<x<y<b$ forces $|ay|>|ax|$ and $|bx|>|by|$ , the cross-ratio is a product of two numbers each greater than $1$ , so it exceeds $1$ exactly when $x\neq y$ , making $d_\Omega$ strictly positive off the diagonal. The additivity along a fixed line is really a statement about ordinary numbers on the real line: cross-ratios computed from the same pair of boundary points $a,b$ telescope multiplicatively as a third collinear point is inserted, so a logarithm turns that telescoping product into a sum — the same phenomenon you can check numerically on any three points of a segment.

Proof

Symmetry and positivity. Write $[a,x,y,b]=\frac{|ay|}{|ax|}\cdot\frac{|bx|}{|by|}$ . Since $a<x<y<b$ on the line, $|ay|=|ax|+|xy|>|ax|$ and $|bx|=|by|+|xy|>|by|$ , so both factors exceed $1$ when $x\ne y$ , hence $[a,x,y,b]>1$ and $d_\Omega(x,y)>0$ ; when $x=y$ both factors equal $1$ . Relabelling $(x,y,a,b)\mapsto(y,x,b,a)$ sends $\frac{|ay||bx|}{|ax||by|}$ to $\frac{|bx||ay|}{|by||ax|}$ , the identical number, so $d_\Omega(x,y)=d_\Omega(y,x)$ .

Additivity on a line. Let $x,y,z$ be collinear with $y$ between $x$ and $z$ , and let $a,b$ be the (common) boundary intersections of that line. Writing all four one-dimensional distances along the line, $[a,x,y,b]\cdot[a,y,z,b]=\frac{|ay|}{|ax|}\frac{|bx|}{|by|}\cdot\frac{|az|}{|ay|}\frac{|by|}{|bz|}=\frac{|az|}{|ax|}\frac{|bx|}{|bz|}=[a,x,z,b],$ since the factors $|ay|$ and $|by|$ cancel. Taking $\frac12\ln$ of both sides gives $d_\Omega(x,y)+d_\Omega(y,z)=d_\Omega(x,z)$ , exactly the equality case of the triangle inequality, so the straight segment $[x,z]$ realizes the Hilbert distance and is a geodesic.

Triangle inequality off a line. For $x,y,z$ not collinear, convexity of $\Omega$ is what is needed: projecting the pair $(x,z)$ through $y$ only ever enlarges the relevant cross-ratio compared with going straight from $x$ to $z$ , because the boundary points seen from $y$ lie no closer to $x$ or $z$ than the boundary points seen directly along $xz$ — this monotonicity, proved carefully using the convexity of $\Omega$ , gives $d_\Omega(x,z)\le d_\Omega(x,y)+d_\Omega(y,z)$ in general (see Hilbert 1895; a full derivation is given in the Handbook of Hilbert Geometry, Ch. 1).

Example: Computing a Hilbert distance from the definition

Let $\Omega$ be the open unit disk $\{x^2+y^2<1\}$ , and let $x=(-0.5,0)$ and $y=(0.5,0)$ . Compute $d_\Omega(x,y)$ directly from the definition $d_\Omega(x,y)=\frac12\ln[a,x,y,b]$ .

Solution

The horizontal line through $x$ and $y$ meets the unit circle at $a=(-1,0)$ and $b=(1,0)$ , in the order $a,x,y,b$ . Along that line, $|ay|=1.5$ , $|bx|=1.5$ , $|ax|=0.5$ , $|by|=0.5$ , so $[a,x,y,b]=\frac{1.5\times1.5}{0.5\times0.5}=\frac{2.25}{0.25}=9$ . Hence $d_\Omega(x,y)=\frac12\ln 9=\ln 3\approx 1.099$ . (Note the pleasant coincidence that $|ay|=|bx|$ here, because $x$ and $y$ are symmetric about the center — that symmetry is what makes the cross-ratio a perfect square, $9=3^2$ .)

The special ellipse case

x^2/a^2+y^2/b^2=1

: here the Hilbert metric

d_\Omega

is not merely analogous to hyperbolic geometry, it is the classical Klein (projective) model of the hyperbolic plane, transported onto this particular ellipse by a linear change of coordinates.

Theorem: The ellipse case is exactly the Klein model of hyperbolic geometry

Let $\mathbb{B}=\{x_1^2+\dots+x_n^2<1\}$ be the open unit ball. Then $(\mathbb{B},d_{\mathbb{B}})$ is isometric to $n$ -dimensional hyperbolic space of constant curvature $-1$ , via the classical Beltrami–Klein model. More generally, if $E\subset\mathbb{R}^n$ is any ellipsoid (the image of $\mathbb{B}$ under an invertible affine map $T$ ), then $(E,d_E)$ is isometric to $(\mathbb{B},d_{\mathbb{B}})$ via $T$ , hence again to hyperbolic space; no other choice of bounded convex $\Omega$ (not affinely equivalent to an ellipsoid) gives a Hilbert geometry isometric to a constant-curvature space.

Why is it true?

Hilbert's cross-ratio construction is literally the same one Arthur Cayley and Felix Klein used decades earlier to build the projective model of hyperbolic geometry inside a conic — Hilbert's 1895 letter to Klein pointed out that convexity of $\Omega$ , not the special quadratic shape of a conic, is what makes the cross-ratio formula define a metric at all. Specializing his general construction back to $\Omega=\mathbb{B}$ therefore reproduces Cayley and Klein's construction exactly, which is precisely why an ellipse is the one shape for which Hilbert geometry is not merely 'hyperbolic-like' but literally hyperbolic.

Proof

Along a diameter. Take $x=0$ and $y=(r,0,\dots,0)$ with $0<r<1$ . The line through them meets $\partial\mathbb{B}$ at $a=(-1,0,\dots,0)$ and $b=(1,0,\dots,0)$ , so $|ay|=1+r$ , $|ax|=1$ , $|bx|=1$ , $|by|=1-r$ , giving $d_{\mathbb{B}}(0,y)=\frac12\ln\frac{1+r}{1-r}=\operatorname{artanh}(r)$ . This matches exactly the standard formula for hyperbolic distance from the center to a point at Euclidean radius $r$ in the Beltrami–Klein model of curvature $-1$ .

Off a diameter. For general $x,y\in\mathbb{B}$ , choose a rotation $R\in O(n)$ (an isometry of the round ball fixing $0$ ) carrying the line through $x,y$ to a coordinate axis; since $R$ preserves $\mathbb{B}$ and maps the four points $a,x,y,b$ to another quadruple in the same cross-ratio, $d_{\mathbb{B}}(x,y)=d_{\mathbb{B}}(Rx,Ry)$ , reducing to the diametral case worked out above after also translating one point to $0$ via a hyperbolic isometry of $\mathbb{B}$ (a Möbius transformation preserving $\mathbb{B}$ , which likewise preserves cross-ratios). Since the formula obtained this way is exactly the Beltrami–Klein distance formula, $(\mathbb{B},d_{\mathbb{B}})$ is isometric to hyperbolic $n$ -space.

General ellipsoids. Let $T:\mathbb{R}^n\to\mathbb{R}^n$ be an invertible linear (or affine) map with $T(\mathbb{B})=E$ . An affine map is a projective transformation of $\mathbb{RP}^n$ that fixes the hyperplane at infinity, and cross-ratios of collinear points are invariant under every projective transformation; since $T$ sends the line through $x,y\in\mathbb{B}$ to the line through $Tx,Ty\in E$ , and sends the boundary pair $a,b$ to the boundary pair $Ta,Tb$ of $E$ , it sends the defining quadruple to the defining quadruple, so $[a,x,y,b]=[Ta,Tx,Ty,Tb]$ and hence $d_E(Tx,Ty)=d_{\mathbb{B}}(x,y)$ for all $x,y$ . Thus $T:(\mathbb{B},d_{\mathbb{B}})\to(E,d_E)$ is an isometry, so $(E,d_E)$ is again hyperbolic $n$ -space.

Example: Affine invariance: the same distance, transported onto an ellipse

Let $E=\{x_1^2/4+x_2^2/1.44<1\}$ be the ellipse with semi-axes $a=2$ , $b=1.2$ , the image of the unit disk $\mathbb{B}$ under the linear map $T(x_1,x_2)=(2x_1,1.2x_2)$ . Using the disk computation from the intuition section ( $x=(0,0)$ , $y=(0.6,0)$ , giving $d_{\mathbb{B}}(x,y)=\ln 2$ ), compute $d_E(Tx,Ty)$ directly from the definition on $E$ , and check it against Theorem 'The ellipse case is exactly the Klein model of hyperbolic geometry'.

Solution

$T$ sends $x=(0,0)$ to $x'=(0,0)$ and $y=(0.6,0)$ to $y'=(1.2,0)$ . The horizontal line through $x',y'$ meets $\partial E$ where $x_1^2/4=1$ , i.e. at $a'=(-2,0)$ and $b'=(2,0)$ . Along that line, $|a'y'|=1.2+2=3.2$ , $|b'x'|=2-0=2$ , $|a'x'|=0-(-2)=2$ , $|b'y'|=2-1.2=0.8$ , so $[a',x',y',b']=\frac{3.2\times 2}{2\times 0.8}=\frac{6.4}{1.6}=4$ , giving $d_E(x',y')=\frac12\ln 4=\ln 2$ . This is exactly $d_{\mathbb{B}}(x,y)=\ln 2$ , confirming the theorem's claim that the linear map $T$ is an isometry from $(\mathbb{B},d_{\mathbb{B}})$ onto $(E,d_E)$ : no new computation was really needed, because $T$ is affine and the cross-ratio never notices affine reparametrizations of the line.

AdvancedBeyond the ellipse: Hilbert's fourth problem and Finsler geometry

Hilbert's 1895 letter to Klein was not an isolated curiosity: it fed directly into Problem IV on his famous list of 23 problems presented at the 1900 International Congress of Mathematicians in Paris, which asks for a characterization of all metrics on a region of real projective space whose geodesics are exactly the ordinary straight line segments. By Theorem 'The Hilbert distance is a genuine metric, with straight segments as geodesics' above, every Hilbert geometry built from a convex domain $\Omega$ is automatically an example. Georg Hamel solved the smooth ('regular') case in 1901, showing that every sufficiently smooth such metric is locally a Minkowski (translation-invariant, norm-induced) metric; the problem without any smoothness assumption was substantially resolved through Herbert Busemann's integral-geometric approach — representing admissible metrics via measures on the space of hyperplanes, in the spirit of the Crofton formula — and Aleksei Pogorelov's 1973 general solution built on it, though the two-dimensional case has continued to attract refinements since. A single fact governs exactly how 'curved' a Hilbert geometry $(\Omega,d_\Omega)$ can be: $d_\Omega$ comes from a genuine Riemannian metric of constant curvature $-1$ precisely when $\Omega$ is an ellipsoid (the theorem above); for every other bounded convex $\Omega$ , $d_\Omega$ is still a perfectly good metric with straight-line geodesics, but it arises from a non-Riemannian Finsler structure — the 'unit ball' of directions at each point is a rescaled copy of $\Omega$ itself rather than a round ellipsoid, so lengths depend on direction in a way no Riemannian metric can reproduce. Infinitesimally, if $t_+(x,v)$ and $t_-(x,v)$ denote how far one can travel from $x\in\Omega$ along $+v$ and $-v$ before exiting $\Omega$ , the Hilbert distance is generated by the Finsler norm below. When $\Omega$ is centrally symmetric ( $\Omega=-\Omega$ , e.g. a square or a regular hexagon centered at the origin), this Finsler structure becomes translation-invariant and reduces, near the center, to the geometry of a normed vector space — a second classical family of symmetric convex domains alongside the ellipsoids, genuinely Riemannian only when that norm happens to be Euclidean.

F_\Omega(x,v) = \frac12\,\|v\|\left(\frac{1}{t_+(x,v)}+\frac{1}{t_-(x,v)}\right)

Common mistake. Three easy mistakes with Hilbert geometry. (1) Assuming any Hilbert geometry 'is' hyperbolic geometry — it only is, exactly, when

\Omega

is an ellipsoid; for a square or a triangle,

d_\Omega

is a perfectly good metric with straight-line geodesics, but it is Finsler, not Riemannian, and its geometry is genuinely different from constant-curvature hyperbolic space. (2) Forgetting the factor

\frac12

d_\Omega(x,y)=\frac12\ln[a,x,y,b]

, or forgetting that the four points must be taken in the order

a,x,y,b

along the line — reordering them inverts the cross-ratio and flips the sign of the logarithm. (3) Confusing the Hilbert metric with its non-symmetric building block, the Funk metric

F_\Omega(x,y)=\ln\frac{|bx|}{|by|}

(using only the far boundary point

b

beyond

y

): the Funk metric already has straight-line geodesics but is not symmetric (

F_\Omega(x,y)\ne F_\Omega(y,x)

in general), and the Hilbert metric is recovered only after explicitly symmetrizing,

d_\Omega(x,y)=\frac12(F_\Omega(x,y)+F_\Omega(y,x))

AdvancedBridges forward: convex projective structures and their limits

Once a compact manifold $M$ is presented as $\Omega/\Gamma$ for a properly convex domain $\Omega\subset\mathbb{RP}^n$ and a discrete group $\Gamma$ of projective transformations acting freely and cocompactly on $\Omega$ , it carries a convex projective structure, and the Hilbert metric $d_\Omega$ descends to a genuine (Finsler, generally non-Riemannian) Riemannian-like metric on $M$ itself. Studying how such structures deform — their curvature, their geodesic flows, their moduli — pulls in the tools of smooth differential geometry (Higgs bundles, harmonic maps, connections) even though the underlying metric is only Finsler; this is one path from Hilbert geometry towards the broader landscape of differential geometry on manifolds. In the opposite direction, letting a strictly convex domain $\Omega_t$ degenerate — its boundary flattening toward a polytope as a parameter $t$ moves to an extreme — makes the Hilbert metric increasingly polyhedral: distances become governed by which facet of the limiting polytope a geodesic runs closest to, in a 'max/min of linear functions' way that echoes, as an analogy rather than a literal identity, how tropical geometry replaces curved algebraic varieties by piecewise-linear polyhedral complexes under logarithmic degeneration. Both directions matter for later chapters of this library: the differential-geometric direction towards higher Teichmüller theory and Anosov representations, and the degenerating, piecewise-linear direction towards tropical geometry.

Historical note

In 1871, Felix Klein had shown that Arthur Cayley's projective distance formula, applied to the interior of a conic, gives a model of hyperbolic geometry in which geodesics are straight lines — the Klein (or Cayley–Klein) model. In a short 1895 letter to Klein, published the same year in Mathematische Annalen, David Hilbert asked what happens if the conic is replaced by an arbitrary bounded convex region: he showed the same cross-ratio recipe still produces a metric with straight-line geodesics, generalizing Klein's construction far beyond conics for the first time. This letter fed directly into Problem IV of the 23 problems Hilbert posed at the 1900 International Congress of Mathematicians, part of the same circle of ideas that shaped his broader program for the foundations of geometry. It is worth contrasting Hilbert's projective route with Henri Poincaré's own, independently discovered conformal models of the hyperbolic plane (the upper half-plane and the open disk with the Poincaré metric, from the early 1880s): Poincaré's models preserve angles but bend geodesics into circular arcs, while Klein's and Hilbert's preserve straight lines but distort angles — two genuinely different faithful pictures of the very same hyperbolic geometry. Hilbert geometry itself lay relatively dormant for decades after Hamel, Busemann, and Pogorelov settled Hilbert's fourth problem, before undergoing a substantial revival from the 2000s onward, when Yves Benoist's sequence of papers on 'convexes divisibles' (divisible convex sets) showed that non-ellipsoidal Hilbert geometries admitting a cocompact group of symmetries are far more abundant and exotic than anyone had expected, reconnecting the subject to geometric group theory and the theory of discrete subgroups of Lie groups.

David Hilbert Henri Poincaré

ResearchConvex divisible domains and higher-rank rigidity

Research frontier as of 2026

Where the subject stands (as of 2026). A properly convex domain $\Omega\subset\mathbb{RP}^n$ is called divisible if some discrete group $\Gamma$ of projective transformations acts on it freely and cocompactly, so $\Omega/\Gamma$ is a closed manifold carrying a convex projective structure. When $\Omega$ is an ellipsoid, this recovers the classical closed hyperbolic manifolds; Yves Benoist's foundational 2000s work ('Convexes divisibles I–IV') constructed the first strictly convex divisible domains that are not ellipsoids in dimension $\ge 3$ , and showed a rigidity dichotomy — such a $\Gamma$ is either Zariski-dense in $\mathrm{PGL}(n+1,\mathbb{R})$ , or $\Omega$ splits as a product or a symmetric cone. Two questions remain genuinely open: a full classification of which properly convex domains are divisible in a given dimension is unknown beyond low dimensions, and the fine regularity of the boundary $\partial\Omega$ of exotic (non-symmetric) strictly convex divisible domains — known to fail to be $C^1$ off a small exceptional set in general — is only partly understood. Since roughly 2015, techniques from Hilbert geometry have become central to the fast-moving theory of Anosov representations and higher Teichmüller theory, which studies discrete, geometrically well-behaved representations of surface (and more general hyperbolic) groups into higher-rank Lie groups; convex projective structures on surfaces (Hilbert geometries with $\Gamma$ a surface group) give some of the founding examples, through Choi–Goldman's identification of Hitchin's component for $\mathrm{PGL}(3,\mathbb{R})$ with the space of such structures. Research programs meeting in 2025 (at ICTS Bengaluru and IHP Paris, among other venues) have pushed the theory further: towards Anosov representations into non-Archimedean and mixed-characteristic groups, towards spectral statistics of Hilbert length spectra for tuples of representations, and towards 'higher-higher' Teichmüller spaces built from pseudo-Riemannian hyperbolic spaces $\mathbb{H}^{p,q}$ — all of it still traceable back to Hilbert's 1895 cross-ratio.

In the unit disk $\Omega=\{x^2+y^2<1\}$ , let $x=(0,0)$ and $y=(0.6,0)$ . The line through them meets the circle at $a=(-1,0)$ and $b=(1,0)$ , giving $|ay|=1.6$ , $|bx|=1$ , $|ax|=1$ , $|by|=0.4$ . What is $d_\Omega(x,y)$ ?

For which shape of bounded convex domain $\Omega$ does the Hilbert metric $d_\Omega$ coincide exactly with the classical Klein (Beltrami–Klein) model of hyperbolic geometry?

Which statement correctly describes the Hilbert metric $d_\Omega$ on a general bounded convex domain $\Omega$ that is not an ellipsoid (say, a square or a triangle)?

Hilbert's fourth problem (1900) asks for a characterization of all metrics on a region of real projective space whose geodesics are exactly the ordinary ___.

References

Athanase Papadopoulos, Marc Troyanov (eds.) (2014). Handbook of Hilbert Geometry · DOI:10.4171/147
Athanase Papadopoulos, Marc Troyanov (2014). From Funk to Hilbert Geometry · arXiv:1406.6983
David Hilbert (1895). Über die gerade Linie als kürzeste Verbindung zweier Punkte · DOI:10.1007/BF02096204