\(L_{p}\)-dual affine surface area forms of Busemann-Petty type problems (Q2344082)

The notion of intersection body, which was introduced by \textit{E. Lutwak} in [Adv. Math. 71, No. 2, 232--261 (1988; Zbl 0657.52002)], is known to be fundamentally connected to the well-known Busemann-Petty problem. Lutwak proved that the Busemann-Petty problem has a positive answer if \(K\) is an intersection body in \(\mathbb R^n\). The \(L^p\)-intersection body was defined by \textit{C. Haberl} and \textit{M. Ludwig} [Int. Math. Res. Not. 2006, No. 17, Article ID 10548, 29 p. (2006; Zbl 1115.52006)]. Later, \textit{J. Yuan} [J. Math. Anal. Appl. 338, No. 2, 1431--1439 (2008; Zbl 1136.52006)] generalized the classical Busemann-Petty problem from intersection bodies to \(L^p\)-intersection bodies. The authors put forward \(L^p\)-dual affine surface area forms of Busemann-Petty type problems and prove a negative form of the Busemann-Petty-type problem for \(L^p\) intersection bodies.