Surface area of Convex bodies contained in one another

Question

Suppose we have two compact convex bodies one contained in the other in $\mathbb{R}^n$, $C\subset D\subset \mathbb{R}^n$. Does it follow that the ($n-1$ dimensional) surface area of $C$ is less than $D$? If so is there a natural sequence of $n-k$ dimensional quantities ($g_0=$Volume, $g_1=$Surface area,...) such that whenever $C\subset D\subset \mathbb{R}^n$ (where $C$ and $D$ are again compact) and $n>k$ we have $g_k(C)>g_k(D)$?