  We prove that if a quasiconvex subset $X$ of a metric space $Y$ has finite
Nagata dimension and is Lipschitz $k$-connected or admits Euclidean
isoperimetric inequalities up to dimension $k$ for some $k$ then $X$ is
isoperimetrically undistorted in $Y$ up to dimension $k+1$. This generalizes
and strengthens a recent result of the third named author and has several
consequences and applications. It yields for example that in spaces of finite
Nagata dimension, Lipschitz connectedness implies Euclidean isoperimetric
inequalities, and Euclidean isoperimetric inequalities imply coning
inequalities. It furthermore allows us to prove an analog of the
Federer-Fleming deformation theorem in spaces of finite Nagata dimension
admitting Euclidean isoperimetric inequalities.
