It is shown that if
B=[-b_1, b_1] \times \cdots \times [-b_n,b_n] \subset {\mathbb{R}}^n,
B
=
[
−
b
1
,
b
1
]
×
⋯
×
[
−
b
n
,
b
n
]
⊂
R
n
,
where
b_i>0
b
i
>
0
for
i=1,...,n\,,
i
=
1
,
.
.
.
,
n
,
and if
A
A
is a convex and compact subset of
B
B
of positive Lebesgue measure, which is preserved by reflections with respect to all coordinate hyperplanes
x_i=0
x
i
=
0
for
i=1,...,n \,,
i
=
1
,
.
.
.
,
n
,
then
A
A
is convexly majorized by
B,
B
,
i.e., for every continuous convex function
v
v
defined over
B,
B
,
the mean of
v
v
over
A
A
is not exceeding the mean of
v
v
over
B.
B
.
In the proof an n-dimensional extension of the integral form of the Chebysev inequality, which was given by L. Vietoris [Eine Verallgemeinerung eines Satzes von Tschebyscheff, Univ. Beograd Publ. Elektrotehn, Fak. Ser. Mat. Fiz 461-497 (1974) 115-117], is used