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(1) Could anyone tell me why it is useful or natural to work with Hilbert metric on the relative interior of some simplex or in general convex set in $\mathbb R^n$?

(2)I do not get the definition of Hilbert metric in the wiki, which says the set ''does not contain a line'', so far I know a convex set always contains a line between any two points on the set.

Myshkin
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(2) Convex sets contain the interval between two points, but not necessarily the whole line. What this condition says is that your set doesn't go to infinity.

(1) As Wikipedia says the prime example are ellipses that define hyperbolic geometry. In particular the distance to the boundary becomes infinite.

mathquest
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