Busemann-Petty-type problem for \(\mu\)-intersection bodies (Q6620606)

The original Busemann-Petty problem asks whether two origin-symmetric convex bodies \(K,L\subset{\mathbb R}^n\) \((n\ge 3\)) with \(\mu_{n-1}(K\cap u^\perp)\le \mu_{n-1}(L \cap u^\perp)\) for all unit vectors \(u\in{\mathbb R}^n\) must satisfy \(\mu_n(K)\le \mu_n(L)\), where \(\mu_k\) denotes \(k\)-dimensional Lebesgue measure. \textit{E. Lutwak} [Adv. Math. 71, No. 2, 232--261 (1988; Zbl 0657.52002)], using dual mixed volumes and radial Blaschke addition, proved that this is true for origin-symmetric star bodies \(K,L\) if \(K\) is an intersection body. The present authors extend this approach to more general measures \(\mu_k\), provided these have a density (with respect to Lebesgue measure) which is non-negative, continuous, even and homogeneous. For this, the notion of intersection body is generalized appropriately. It is also shown that the answer in the general case is negative if \(K\) is not centrally symmetric.