Lipschitz Smoothings of Polyhedral Manifolds

We use a recent result of C. Lange to obtain a converse to a theorem of B. Bowditch in dimension at most $4$. In particular, we show that, for $n \leq 4$, a polyhedral $n$-manifold $X$ with bounded geometry is $K$-bi-Lipschitz homeomorphic to a Riemannian manifold $M$. We bound the constant $K$, the curvature, and the injectivity radius of $M$ by the bounds on the geometry of $X$.