Complex intersection bodies (Q2854254)

The complex convex bodies and complex star bodies in \({\mathbb C}^n\) considered in this paper can be viewed as origin symmetric convex or star bodies \(K\) in \({\mathbb R}^{2n}\) with the property that the Minkowski functional \(\|\cdot\|_K\) satisfies NEWLINE\[NEWLINE \|\xi\|_K =\|R_\theta(\xi_{11}, \xi_{12}),\dots,R_\theta(\xi_{n1},\xi_{n2})\|_KNEWLINE\]NEWLINE for all \(\xi=(\xi_{11}, \xi_{12},\dots,\xi_{n1},\xi_{n2})\in{\mathbb R}^{2n}\) and all \(\theta\in[0,2\pi]\), where \(R_\theta\) denotes the rotation of \({\mathbb R}^2\) by the angle \(\theta\). In analogy to the real intersection bodies, the authors introduce complex intersection bodies of complex star bodies, and they are able to find complex analogues of many facts known about real intersection bodies. After extending the Fourier transform techniques that were introduced by Koldobsky in the field of real intersection bodies, it is shown that an origin symmetric complex star body in \({\mathbb R}^{2n}\) is a complex intersection body if and only if the function \(\|\cdot\|^{-2}\) represents a positive-definite distribution. As a consequence, complex intersection bodies can be interpreted as real \(2\)-intersection bodies and also as generalized 2-intersection bodies in \({\mathbb R}^{2n}\). In analogy to Busemann's convexity theorem, it is proved that complex intersection bodies of symmetric complex convex bodies are convex. Further results deal with stability in the complex Busemann--Petty problem for arbitrary measures and with related hyperplane inequalities.