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.