On the growth of Lebesgue constants for convex polyhedra

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 $W$.