Math 2.0

Geometry Definition

Hilbert metric

Builds on Convex setEuclidean distanceMetric space

Definition

Let n≥1n \ge 1 and let Ω⊆ℝn\Omega \subseteq \R^n be a bounded open convex set. Write ∣pq∣\abs{pq} for the Euclidean distance of points pp and qq. Let x,y∈Ωx, y \in \Omega be distinct. The line through xx and yy meets the boundary of Ω\Omega in exactly two points aa and bb. Name them so that a,x,y,ba, x, y, b lie on the line in this order. The Hilbert metric on Ω\Omega is dΩ(x,y)=12log⁡∣ay∣ ∣xb∣∣ax∣ ∣yb∣,d_\Omega(x, y) = \frac{1}{2} \log \frac{\abs{ay} \, \abs{xb}}{\abs{ax} \, \abs{yb}}, with dΩ(x,x)=0d_\Omega(x, x) = 0.

Remarks

The map dΩd_\Omega is a metric on Ω\Omega; the triangle inequality uses the convexity of Ω\Omega.

Example: let Ω=(−1,1)⊂ℝ\Omega = (-1, 1) \subset \R and 0<y<10 < y < 1. For x=0x = 0 the boundary points are a=−1a = -1 and b=1b = 1, so dΩ(0,y)=12log⁡1+y1−yd_\Omega(0, y) = \frac{1}{2} \log \frac{1 + y}{1 - y}, which tends to ∞\infty as y→1y \to 1.

Non-example: the open half-plane {(u,v)∈ℝ2:v>0}\set{(u, v) \in \R^2 : v > 0} is convex but not bounded. A horizontal line in it does not meet the boundary at all, so the formula has no meaning.

Some authors omit the factor 12\frac{1}{2}. When Ω\Omega is the open unit ball, dΩd_\Omega is the hyperbolic distance of the Beltrami–Klein model.

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To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Convex set, concept, no evidence levelConvex setconceptElement, definition, no evidence levelElementdefinitionEuclidean distance, concept, no evidence levelEuclidean distanceconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionHilbert metric, definition, no evidence levelHilbert metricdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Convex set concept
  2. Element definition
  3. Euclidean distance concept
  4. Proposition concept
  5. Real numbers object
  6. Set definition
  7. Implication definition
  8. Predicate definition
  9. Ordered pair definition
  10. Existential quantifier definition
  11. Universal quantifier definition
  12. Cartesian product definition
  13. Subset definition
  14. Uniqueness quantifier definition
  15. Binary relation definition
  16. Function definition
  17. Metric space definition
  18. Hilbert metric definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert_metric
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TagHDD0EB48AB
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Kinddefinition
WritingW0 stub
MSC51F99, 52A20, 53C60
DefinesHilbert metric
UsesConvex set, Euclidean distance, Metric space

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