The modified complex Busemann-Petty problem on sections of convex bodies (Q1041499)

The author gives necessary conditions on the complex section function \( S_{CD} (\xi) = Vol_{2n-2} (D \cap H_{\xi}) \), where \( D \subset {\mathbb C}^n = {\mathbb R}^{2n} \) is an origin symmetric convex body, \( H_{\xi} \subset {\mathbb C}^{n} \) is a complex hyperplane, \( H_{\xi} \bot \xi \in {\mathbb C}^{n} \), \( | \xi | = 1 \), for the affirmative answer on the Busemann-Petty problem in a complex Euclidean space \( {\mathbb C}^{n} \) of any dimension \( n \geq 3 \). These conditions are formulated in terms of fractional powers \( \Delta^{\alpha/2}_{\xi} \) of the Laplace operator applied to the extension of \( S_{CD} (\xi) \) to the ambient space \( {\mathbb R}^{2n} \) as a homogeneous function of degree \(-2\). Sufficient conditions for the negative answer are given as well. Similar result in the case of real Euclidean spaces was obtained by \textit{A. Koldobsky, V. Yaskin} and \textit{M. Yaskina} [Isr. J. Math. 154, 191--207 (2006; Zbl 1139.52009)].