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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?

Thanks,

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  • $\begingroup$ Could you be more precise about what you want? In general two bodies with the same volume have different values of $W_i$. $\endgroup$ – Deane Yang Feb 20 '15 at 22:26
  • $\begingroup$ @DeaneYang: Thanks for your comment. In fact, there is an assumption that $K\subset L$. However, I have a little doubt about my question. I am not sure that $W_i(K) \leq W_i(L)$ if $K\subset L$. The following question maybe is more precise. Given convex bodies $K,L,M$ such that $L\subset M$. Does there exist the following inequality $W_i(K,M)-W_i(K,L) \geq c_n W_i(K)^{\frac{n-i-1}{n-i}} (V(M)^{\frac 1n} -V(L)^{\frac 1n})$, $0\leq i < n$ with $c_n$ is a positive constant (maybe depends on $n$), here $W_i(K,L)$ is mixed Quermassintegral. Thanks $\endgroup$ – nguyen0610 Feb 21 '15 at 12:54

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