Article

Israel Journal of Mathematics

, Volume 135, Issue 1, pp 157-179

Some inequalities about mixed volumes

  • M. FradeliziAffiliated withEquipe d’Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée Email author 
  • , M. MeyerAffiliated withEquipe d’Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée
  • , A. GiannopoulosAffiliated withDepartment of Mathematics, University of Crete

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Abstract

We prove inequalities about the quermassintegralsV k (K) of a convex bodyK in ℝ n (here,V k (K) is the mixed volumeV((K, k), (B n ,n − k)) whereB n 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 ℝ n if and only ifk=2 ork=1. (ii) Let 0≤kpn. Then, for everyp-dimensional subspaceE of ℝ n ,
$$\frac{{V_{n - k} \left( K \right)}}{{\left| K \right|}} \geqslant \frac{1}{{\left( {_{n - p}^{n - p + k} } \right)}}\frac{{V_{p - k} \left( {P_E K} \right)}}{{\left| {P_E K} \right|}},$$
whereP E K denotes the orthogonal projection ofK ontoE. The proof is based on a sharp upper estimate for the volume ratio |K|/|L| in terms ofV n−k (K)/V n−k (L), wheneverL andK are two convex bodies in ℝ n such thatKL.