Let $K$ be a convex body (a compact, convex subset of $R^n$ with empty interior), denote $W_0(K),W_1(K),\cdots,W_n(K)$ the Quermassintegrals of $K$. Note that $W_0(K)=V(K)$ the volume of $K$, and $W_n(K)$ is volume of the Euclidean unit ball. It is known that if $K\subset L$ are convex bodies, then $W_i(K) \leq W_i(L)$. My question is that do we have an estimate from below for $W_i(L)-W_i(K)$ with $i< n$ in terms of $V(L)-V(K)$? The case $i=0$ is evident.
Can anyone help me answer this question or give a reference for it?