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 \(W\).