The classical Busemann-Feller lemma in Euclidean space says the following.


Let $K\subset \mathbb{R}^n$ be a closed convex set.  

Then


1. for any point $x\in \mathbb{R}^n$ there exists unique nearest point in $K$; let us denote it by $p(x)$.
2. the map $p\colon \mathbb{R}^n \to K$ does not increase distances, i.e. is 1-Lipschitz.


**Question. Does this result hold in an $n$-dimensional hyperbolic space? A reference would be helpful.**


Remark. I think part (1) is true for hyperbolic and spherical spaces; in the spherical space one should require in addition that $K$ is contained in an open half-sphere. I think part (2) is not true in the spherical space.



Busemann-Feller lemma in hyperbolic space
In any Hadamard space, projection into convex sets is non-expansive; see Proposition 2.4(4) in *[Metric spaces of non-positive curvature](https://www.springer.com/gp/book/9783540643241)* by Bridson and Haefliger.


