Monotonicity of surface area and diameter bounds for convex bodies

Let $A, B$ be convex compact sets in $\mathbb {R}^{3}$, with $A \subset B$. Prove that the surface area of the boundary of the set $A$ is no greater than the surface area of the boundary of the set $B$.

Let $T$ be an arbitrary convex compact body in $\mathbb {R}^{m}$, $V=\lambda_{m}(T)$. Prove that if the $(m-1)$-dimensional volume of the projection of $T$ onto any hyperplane is not less than $S$, then $\operatorname {diam} T \leqslant m V / S$.