Israel Journal of Mathematics

, Volume 135, Issue 1, pp 157–179

# Some inequalities about mixed volumes

Article

## Abstract

We prove inequalities about the quermassintegralsVk(K) of a convex bodyK in ℝn (here,Vk(K) is the mixed volumeV((K, k), (Bn,n − k)) whereBn is the Euclidean unit ball). (i) The inequality
$$\frac{{V_k \left( {K + L} \right)}}{{V_{k - 1} \left( {K + L} \right)}} \geqslant \frac{{V_k \left( K \right)}}{{V_{k - 1} \left( K \right)}} + \frac{{V_k \left( L \right)}}{{V_{k - 1} \left( L \right)}}$$
holds for every pair of convex bodiesK andL in ℝn if and only ifk=2 ork=1. (ii) Let 0≤kpn. Then, for everyp-dimensional subspaceE of ℝn,
$$\frac{{V_{n - k} \left( K \right)}}{{\left| K \right|}} \geqslant \frac{1}{{\left( {_{n - p}^{n - p + k} } \right)}}\frac{{V_{p - k} \left( {P_E K} \right)}}{{\left| {P_E K} \right|}},$$
wherePEK denotes the orthogonal projection ofK ontoE. The proof is based on a sharp upper estimate for the volume ratio |K|/|L| in terms ofVn−k(K)/Vn−k(L), wheneverL andK are two convex bodies in ℝn such thatKL.

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