Volume differences of mixed complex projection bodies (Q464225)

If \(K\) is a convex body in \({\mathbb R}^n\), its projection body \(\Pi K\) is the convex body whose support function is given by \(h(\Pi K,u)=(n/2)V\bigl(K[n-1],[-u,u]\bigr)\), where \(V\bigl(K[n-1],[-u,u]\bigr)\) denotes the mixed volume of \(n-1\) copies of \(K\) and the segment \([-u,u]\).NEWLINENEWLINEIn [Adv. Math. 227, No. 2, 830--846 (2011; Zbl 1217.52009)], \textit{J. Abardia} and \textit{A. Bernig} studied projection bodies in a complex vector space: for convex bodies \(K\) in \({\mathbb C}^n\) and \(C\) in \({\mathbb C}\), the projection body of \(K\) associated to \(C\) is defined by \(h(\Pi_CK,u)=V\bigl(K[2n-1],C\cdot u\bigr)\), where \(C\cdot u=\{cu:c\in C\subset{\mathbb C}\}\). They proved that any continuous, translation-invariant Minkowski valuation, which is contravariant under the complex special linear group, is the projection body operator \(\Pi_C\), for some convex body \(C\subset{\mathbb C}\). The version of this result over \({\mathbb R}\) was proved by \textit{M. Ludwig} [Trans. Am. Math. Soc. 357, No. 10, 4191--4213 (2005; Zbl 1077.52005)]. In both the real and the complex cases, geometric inequalities like Brunn-Minkowski or Aleksandrov-Fenchel inequalities were stated.NEWLINENEWLINEIn the paper under review, the author obtains inequalities for the so-called volume difference involving \(\Pi_C\), following the proofs of the analogous results in the real setting.