Thompson Metric in Order Cones

• The Thompson metric is an order-theoretic projective metric defined on proper cones, measuring distances through logarithms of scaling factors.
• It underpins fixed-point theory and contraction analyses in spaces like symmetric positive-definite matrices and convex domains.
• Its Finsler structure and explicit geodesic formulas bridge geometric and norm-based approaches, facilitating numerical and analytical methods.

The Thompson metric is a fundamental projective metric defined on ordered vector spaces equipped with proper cones, serving as a canonical Finsler geometry for cones of positive elements in analysis, geometry, optimization, and operator theory. Unlike Riemannian metrics, the Thompson metric arises exclusively from the order structure of the cone and is intimately connected to homogeneity, monotonicity, and projective invariance. Its explicit forms on cones of symmetric positive-definite matrices, spectral densities, and convex domains make it an indispensable tool for fixed-point theory, contraction analysis of flows, and geometric approaches to optimization.

1. Formal Definition and Fundamental Properties

Let $X$ be a real vector space and $K \subset X$ a convex cone: $K + K \subset K$, $t K \subset K$ for $t \geq 0$, $K \cap (-K) = \{0\}$. The cone induces a partial order $x \leq y \iff y - x \in K$. On each equivalence class (component) of $K$, two points $x, y$ are linked if there exist scalars $\lambda, \mu > 0$ with $x \leq \lambda y$ and $y \leq \mu x$.

The projective gauge is defined by

$$M(x / y) := \inf \{ \lambda > 0 : x \leq \lambda y \}$$

for $x, y$ in the same component. The Thompson metric on components of $K$ is then

$$d_T(x, y) := \ln \max \bigl\{ M(x / y), M(y / x) \bigr\}$$

or equivalently

$$d_T(x, y) = \inf \left\{ s \in \mathbb{R} : e^{-s} x \leq y \leq e^{s} x \right\}.$$

This metric is always symmetric, satisfies the triangle inequality, and $d_T(x, y) = 0$ if and only if $x = y$ in almost-Archimedean cones (in particular, if $K$ is Archimedean) (Cobzaş et al., 2013).

Key properties:

• Scale-invariance: $d_T(\alpha x, \alpha y) = d_T(x, y)$ for $\alpha > 0$.
• Monotonicity: If $x \leq x'$, $y \leq y'$, then $d_T(x, y) \leq d_T(x', y')$.
• Quasi-convexity: For any $v \in K$, $$d_T((1-t)x + t y, v) \leq \max\{ d_T(x, v), d_T(y, v) \} \; \forall t \in [0, 1]$$.
• Topology: The $d_T$-balls are order-symmetric intervals $$B_T[x, r] = \{ z \in K : e^{-r} x \leq z \leq e^{r} x \}$$ and define a topology that coincides with that of the order-unit norm on suitable subspaces.
• Completeness: For a Banach space $X$ and closed cone $K$, $(K, d_T)$ is complete if and only if $K$ is normal (i.e., order-intervals are norm-bounded) (Cobzaş et al., 2013, Lins, 2023).

On the cone $\mathbb{P}_d$ of $d \times d$ symmetric positive-definite matrices, this specializes to

$$d_T(X, Y) = \max\bigl\{ \log \lambda_{\max}(X^{-1} Y), \log \lambda_{\max}(Y^{-1} X) \bigr\}$$

where $\lambda_{\max}(\cdot)$ denotes the maximal eigenvalue (Weber et al., 2022, Goffrier et al., 2020).

2. Finsler Structure, Geodesics, and Uniqueness

The Thompson metric equips the cone’s interior with a Finsler (but typically not Riemannian) structure. The tangent space norm at $x \in \mathring K$ is

$$\| v \|^T_x := \inf\{ \alpha > 0 : -\alpha x \leq v \leq \alpha x \}$$

This structure makes the cone a (generally) non-strictly convex Finsler manifold (Baggio et al., 2017, Lemmens et al., 2016). The metric is not strictly locally uniquely geodesic except in the trivial one-dimensional case (Bosché, 2012).

Explicit geodesics:

• For finite-dimensional cones, the “projective straight-line” geodesic:

$$\gamma(\tau) = \frac{\sinh(r(1-\tau))}{\sinh r} x + \frac{\sinh(r\tau)}{\sinh r} y$$

connects $x$ and $y$ along a $d_T$-geodesic of length $r = d_T(x, y)$ (Cobzaş et al., 2013).

• On $\mathbb{P}_d$, the affine-invariant geodesic is

$$\gamma(t) = X^{1/2}\big(X^{-1/2} Y X^{-1/2}\big)^t X^{1/2}$$

and every pair $(X, Y)$ can be joined by such a constant-speed geodesic (Bosché, 2012, Goffrier et al., 2020).

Criteria for uniqueness: In the case of positive matrices, a geodesic between $X, Y$ is unique iff the spectrum of $X^{-1/2} Y X^{-1/2}$ is $\{\beta^{-1}, \beta\}$ for some $\beta > 1$ (Lemmens et al., 2013). For general cones, uniqueness corresponds to the absence of additional dimensions in which the geodesic can deviate (“tilting” criterion).

3. Thompson Metric on Concrete Cones and Domains

Symmetric Cones

A symmetric cone $\mathcal{C}$, the interior of squares in a Euclidean Jordan algebra $J$, supports the metric

$d_{\text{Thom}}(a, b) = \| \log P(a^{-1/2}) b \|$

where $P(x)$ is the quadratic representation, and the norm is the spectral norm (maximum modulus of eigenvalues). For positive-definite matrices this reduces to the operator norm of the matrix logarithm (Bosché, 2012).

Spectral Densities

On the cone of rational spectral densities, the Thompson metric is given by

$$d_T(\Phi_1, \Phi_2) = \log \max \bigl\{ \| W_2^{-1} W_1 \|_{H_\infty}^2, \| W_1^{-1} W_2 \|_{H_\infty}^2 \bigr\}$$

where $W_i$ are minimum-phase spectral factors (Baggio et al., 2017).

Convex Domains

On a bounded convex domain $D \subset \mathbb{R}^n$, define

$$m(p, q) = \inf\{ \lambda > 0 : D - q \subset \lambda (D - p) \}$$

and then $d_T(p, q) = \max\{ \log m(p, q), \log m(q, p) \}$, which is up to a constant equivalent to the Hilbert metric $d_H$ (Serre, 2017).

Polygonal Geometries

In convex polygons $P \subset \mathbb{R}^2$, $d_T$ is realized as the max of the logarithms of scaling factors to boundary intersections (Funk and reverse Funk metrics), with explicit $O(m)$ algorithms for distance and ball construction (Banerjee et al., 3 Mar 2025).

4. Isometries, Embeddings, and Symmetry Results

On symmetric cones, Thompson isometries are described as follows: every isometry is a composition of a Jordan-algebra automorphism and, possibly, a central involution (block inversion in the direct product decomposition) (Bosché, 2012, Lemmens et al., 2016). For the positive cone in a unital $C^*$-algebra, isometries are generated by $\ast$-automorphisms, $\ast$-antiautomorphisms, conjugation, and inversion.

On strictly convex cones of dimension at least $3$, any Thompson isometry is projectively linear: $f(x) = \lambda_x T(x)$ for a linear automorphism $T \in \text{Aut}(C)$ and positive scalar function $\lambda_x$ (Lemmens et al., 2013).

The metric space $(C^\circ, d_C)$ admits an isometric embedding into a finite-dimensional normed space iff $C$ is a simplicial cone; if quasi-isometric, then $C$ must be polyhedral.

5. Fixed-Point Theory, Contraction, and Optimization

Order-preserving, subhomogeneous maps on the interior of a normal cone are nonexpansive in the Thompson metric. Krasnoselskii and Picard iterations for such maps converge to fixed points under mild compactness and spectral (Collatz–Wielandt) conditions (Lins, 2023). For analytic maps, uniqueness and global convergence of iterates are obtained under strict bounds on Collatz–Wielandt numbers.

For matrix equations, in particular Riccati iterations and Brascamp–Lieb fixed-point formulas, analysis in the Thompson metric produces geometric decay and complexity guarantees. For instance, in the regularized Brascamp–Lieb Picard iteration, strict $d_T$-contractivity yields geometric convergence, and Snyder’s inequalities facilitate translation of $d_T$ convergence to Schatten norm convergence (Weber et al., 2022, Snyder, 2016).

For order-preserving flows on cones, the sharp contraction rate in $d_T$ is given by

$$\alpha = -\sup_{t \geq 0, x \in C_0} M\left( D_x \phi(t,x) x - \phi(t,x) / x \right)$$

and is strictly positive only on bounded order-intervals unless the flow is globally strictly contractive (Gaubert et al., 2012).

6. Applications, Midpoints, and Computational Tools

Clustering and Data Geometry

The Thompson metric underlies inductive midrange centroids and cluster assignments for SPD matrix clustering, with fast algorithms based on extremal eigenvalue computations and geometric midpoints (Goffrier et al., 2020).

PDEs and Elliptic Regularity

In singular elliptic boundary-value problems on convex domains, global Lipschitz estimates of logarithms of positive solutions with respect to $d_T$ are essential in establishing existence, uniqueness, and regularity (Serre, 2017).

Metric Geometry and Software

Polygonal Thomspon balls and distances can be visualized and computed efficiently, with open-source software available for exploring Funk, Hilbert, and Thompson geometries in convex polygons (Banerjee et al., 3 Mar 2025).

Midpoints and Metric Geometry

On symmetric cones, the affine span of $d_T$ midpoints between $x, y$ is characterized as a translation of the Peirce zero-space for a special idempotent associated with the boundary spectrum of $x$ with respect to $y$. Explicit formulas for the dimension of the midpoint set are available in simple Euclidean Jordan algebras (Lemmens et al., 2015).

7. Connections to Norms and Operator Theory

Although Thompson metric bounds can seem abstract, explicit inequalities relate $d_T$ to Schatten and Frobenius norms. Given matrices $X,Y$ and $d_T(X, Y) = d$,

$$\| X - Y \|_p \leq \left( \frac{2 (e^d - 1)^p}{e^d} \right)^{1/p} \max\{ \| X \|_p, \| Y \|_p \}$$

with a sharper bound for the Frobenius norm (Snyder, 2016), providing a practical bridge between geometric and classical error control in matrix approximation.

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The Thompson metric thus provides a canonical, order-theoretic geometric framework for the analysis of cones in real vector spaces, Banach spaces, and Jordan algebras, with broad applications in analysis, optimization, geometry, and numerical methods. Its explicit formulation, contractive properties, and projective invariance enable deep analytic control in fixed-point theory, numerical linear algebra, and the geometry of ordered structures.