Geometry Foundations and axiom systems Problem

Hilbert's fourth problem

Problem

Let $U$ be an open convex subset of $\R^n$ . Determine all metrics $d$ on $U$ for which every straight line segment in $U$ is a shortest path. That is, determine all $d$ such that $d(x, z) = d(x, y) + d(y, z)$ whenever $y$ lies on the segment from $x$ to $z$ .

Remarks

Such metrics are called projective metrics, and their geodesics include the straight lines. Examples are the Euclidean metric and the metrics given by any norm on $\R^n$ . Another example is the Hilbert metric on a bounded convex domain; when the domain is a disc, it gives the Beltrami–Klein model of the hyperbolic plane. Hilbert posed the question in his list of problems of 1900. Herbert Busemann gave a construction of such metrics from measures on the space of hyperplanes. Aleksei Pogorelov proved that in the plane every continuous projective metric arises in this way.

Reading path

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Geodesic concept
Line concept
Metric space concept
Hilbert's fourth problem problem

Tag	`XA35DT40XS`
URI	`m2://XA35DT40XS`
Kind	problem
Writing	W0 stub
MSC	51F99, 53C60, 51K05
Status	This copy of the site has no build data. Run `m2 status XA35DT40XS`.
About	Metric space
Uses	Metric space, Line, Geodesic
Same as	Wikidata Q4515035 Wikipedia Hilbert's fourth problem

Hilbert's fourth problem

Problem

Remarks

Reading path

Further reading

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