Monotonicity of perimeter of convex subsets of hyperbolic plane

Question

I think that the perimeter of convex compact sets on the hyperbolic plane is monotone with respect to inclusion.

I am looking for a reference to the above fact.

Answer 1

It follows from the fact that the closest-point projection is short (=distance nonexpanding). Check the solution of 9.47 in our book, altho it is a bit of an overkill.

Answer 2

Does not an elementary Euclidean proof work verbatim? If both sets are polygons, the interior one is obtained from the exterior one by cuts (i.e. transformations $P\to P\cap H$, where $P$ is a polygon, $H$ is a half-plane), after each cut perimeter decreases by the triangle inequality). A general case follows by polygonal approximations.