  We study the broken ray transform on $n$-dimensional Euclidean domains where
the reflecting parts of the boundary are flat and establish injectivity and
stability under certain conditions. Given a subset $E$ of the boundary
$\partial \Omega$ such that $\partial \Omega \setminus E$ is itself flat
(contained in a union of hyperplanes), we measure the attenuation of all broken
rays starting and ending at $E$ with the standard optical reflection rule
applied to $\partial \Omega \setminus E$. By localizing the measurement
operator around broken rays which reflect off a fixed sequence of flat
hyperplanes, we can apply the analytic microlocal approach of Frigyik,
Stefanov, and Uhlmann for the ordinary ray transform by means of a local path
unfolding. This generalizes the author's previous result for the square,
although we can no longer treat reflections from corner points. Similar to the
result for the two dimensional square, we show that the normal operator is a
classical pseudo differential operator of order -1 plus a smoothing term with
$C_{0}^{\infty}$ Schwartz kernel.
