In this paper, new estimates of the Lebesgue constant
L
(
W
)
=
1
(
2
π
)
d
∫
T
d
|
∑
k
∈
W
∩
Z
d
e
i
(
k
,
x
)
|
d
x
\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
⊂
R
d
W\subset {\Bbb R}^d
are obtained. The main result states that if
W
W
is a convex polyhedron such that
[
0
,
m
1
]
×
⋯
×
[
0
,
m
d
]
⊂
W
⊂
[
0
,
n
1
]
×
⋯
×
[
0
,
n
d
]
[0,m_1]\times \dots \times [0,m_d]\subset W\subset [0,n_1]\times \dots \times [0,n_d]
, then
c
(
d
)
∏
j
=
1
d
log
(
m
j
+
1
)
≤
L
(
W
)
≤
C
(
d
)
s
∏
j
=
1
d
log
(
n
j
+
1
)
,
\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
s
is a size of the triangulation of
W
W
.