  In the paper, new estimates of the Lebesgue constant $$
\mathcal{L}(W)=\frac1{(2\pi)^d}\int_{\mathbb{T}^d}\bigg|\sum_{{k}\in W\cap
\mathbb{Z}^d} e^{i({k},\,{x})}\bigg| {\rm d}{ x} $$ for convex polyhedra
$W\subset\mathbb{R}^d$ 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 $$ c(d)\prod_{j=1}^d \log(m_j+1)\le
\mathcal{L}(W)\le C(d)s\prod_{j=1}^d \log(n_j+1), $$ where $s$ is a size of the
triangulation of $W$.
