General complex \(L_p\) projection bodies and complex \(L_p\) mixed projection bodies (Q2135061)

Projection bodies of compact convex bodies in a complex vector space were defined in [\textit{J. Abardia} and \textit{A. Bernig}, Adv. Math. 227, No. 2, 830--846 (2011; Zbl 1217.52009)]. They are parametrized by a compact convex body \(C \subset \mathbb C\). An \(L^p\)-version of complex projection bodies is the so called asymmetric complex \(L^p\) projection body \(\Pi_{p,C}^+\) from [\textit{W. Wang} and \textit{L. Liu}, Adv. Appl. Math. 122, Article ID 102108, 26 p. (2021; Zbl 1458.52003)], which depends on \(C\) and a parameter \(p \geq 1\). By taking a suitable Minkowski linear combination of \(\Pi_{p,C}^+(K)\) and \(\Pi_{p,C}^-(K):=\Pi_{p,C}^+(-K)\), the authors define the general complex \(L^p\) projection body \(\Pi_{p,C}^\lambda(K)\), where \(-1 \leq \lambda \leq 1\). Using standard arguments, they then prove Brunn-Minkowski type and Alexandrov-Fenchel type inequalities for \(\Pi_{p,C}^\lambda\).