On a generalization of Busemann's intersection inequality (Q6585644)

Let \(K\) be an origin-symmetric star body in \({\mathbb R}^n\) (with unit sphere \(S^{n-1}\)), \(\|\cdot\|_K\) its Minkowski functional, and let \(0<p<n\). The author asks whether \N\[\N\int_{S^{n-1}} |(\|\cdot\|_K^{-n+p})^\wedge(\theta)|^{\frac{n}{p}}\,d\theta \le c_{n,p}(\mathrm{vol}_n(K))^{\frac{n}{p}-1}\N\]\Nholds if \(p<n/2\), where \((\cdot)^\wedge\) denotes the Fourier transform and the constant \(c_{n,p}\) is such that equality holds if \(K\) is a ball. It is also asked whether a similar inequality, with the inequality sign reversed, holds for \(p>n/2\), and whether equality holds only if \(K\) is a centered ellipsoid. For \(p=1\), the answer is affirmative, due to Busemann's intersection inequality and a transformation. The general question is motivated by Koldobsky's \(k\)-intersection bodies and a related conjectured inequality. The author gives here an affirmative answer for all origin-symmetric star bodies that are sufficiently close to the Euclidean ball in the Banach-Mazur distance, and he also proves a weaker version of the inequality for all origin-symmetric star bodies. The delicate proofs make use of the Fourier transform of distributions, spherical harmonics, and Stein's interpolation theorem.