Complex \(L_p\) affine isoperimetric inequalities (Q2221778)

Projection body is an important object in convex geometry and the basic inequality involving it is the well-known Petty's projection inequality (also known as the affine isoperimetric inequality). The \(L_p\) (where \(p\geq1\)) extension of Petty's projection inequality was established by \textit{E. Lutwak} et al. [J. Differ. Geom. 56, No. 1, 111--132 (2000; Zbl 1034.52009)]. A recent development in this topic is the work of \textit{C. Haberl} [Calc. Var. Partial Differ. Equ. 58, No. 5, Paper No. 169, 22 p. (2019; Zbl 1425.52009)] where he defined the complex projection body and established the complex affine isoperimetric inequality. In the present paper, the authors extend Haberl to the \(L_p\) version.