Let T be a compact convex region in an n-dimensional Riemannian space,
k
s
{k_s}
be the minimum sectional curvature in T, and
κ
>
0
\kappa > 0
be the minimum normal curvature of the boundary of T. Denote by
P
ν
(
ξ
)
{P^\nu }(\xi )
a v-dimensional sphere, plane or hyperbolic plane of curvature
ξ
\xi
. We assume that
k
s
{k_s}
, k are such that on
P
2
(
k
s
)
{P^2}({k_s})
there exists a circumference of curvature k. Let
R
0
=
R
0
(
κ
,
k
s
)
{R_0} = {R_0}(\kappa ,{k_s})
be its radius. Now, let Q be a convex (in interior sense) m-dimensional surface in T whose normal curvatures with respect to any normal are not greater than x satisfying
0
⩽
χ
>
κ
0 \leqslant \chi > \kappa
. Denote by
L
χ
{L_\chi }
the length of a circular arc of curvature x in
P
2
(
k
s
)
{P^2}({k_s})
with the distance
2
R
0
2{R_0}
between its ends. We prove that the volume of Q does not exceed the volume of a ball in
P
m
(
k
s
−
(
n
−
m
)
χ
2
)
{P^m}({k_s} - (n - m){\chi ^2})
of radius
1
2
L
χ
\tfrac {1}{2}{L_\chi }
. These volumes are equal when T is a ball in
P
n
(
k
s
)
{P^n}({k_s})
and Q is its m-dimensional diameter.