On the growth of Lebesgue constants for convex polyhedra

In this paper, new estimates of the Lebesgue constant \begin{equation*} \mathcal {L}(W)=\frac 1{(2\pi )^d}\int _{{\Bbb T}^d}\bigg |\sum _{\mathbfit {k}\in W\cap {\Bbb Z}^d} e^{i(\mathbfit {k}, \mathbfit {x})}\bigg | \textrm {d}\mathbfit {x} \end{equation*} for convex polyhedra $W\subset {\Bbb R}^d$ are obtained. The main result states that if $W$ is a convex polyhedron such that $[0,m_1]\times \dots \times [0,m_d]\subset W\subset [0,n_1]\times \dots \times [0,n_d]$, then \begin{equation*} c(d)\prod _{j=1}^d \log (m_j+1)\le \mathcal {L}(W)\le C(d)s\prod _{j=1}^d \log (n_j+1), \end{equation*} where $s$ is a size of the triangulation of $W$.