Let \(V\) be a regular subvariety of a non-degenerate analytic polyhedron \(\Omega\subset\mathbb{C}^n\). If \(V\) intersects \(\partial\Omega\) transversally in a certain sense, then each bounded holomorphic function on \(V\) has a bounded holomorphic extension to \(\Omega\). Furthermore, a function in \(H^p(V)\) has an extension in \(H^p(\Omega)\). Under a weaker transversality condition each \(f\in{\mathcal O}(V)\cap L^p(V)\) has an extension to a function on \({\mathcal O}(\Omega)\cap L^p(\Omega)\), \(p<\infty\).