The main result of the authors is the following: For any convex polyhedron \(W\) in \(\mathbb R^n\), \(p\in(1,\infty)\), and \(N\in\mathbb N\), there are constants \(\gamma_1(W,p,n)\) and \(\gamma_2(W,p,n)\) such that \[ \gamma_1N^{n(p-1)}\leq \int_{\mathbb T^n}\Bigl|\sum_{k\in NW} e^{ik\cdot x}\Bigr|^p \,dx\leq \gamma_2N^{n(p-1)}. \] This is an extension of known fact in the case \(n=1,2\) and general \(n\) with \(p=1\), and they give a new proof in the case \(n=2\). They also discuss general set \(B\) with non-empty interior, and give upper and lower estimates. If \(n=1\), \(\Bigl\|\sum_{k=-N}^{N}e^{ikx}\Bigr\|_{L^1(\mathbb T)} (\sim \frac{4}{\pi^2}\log N)\) is the familiar Lebesgue constant.