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Description

The eighth Busemann-Petty problem asks the following question: If for an origin-symmetric convex body $K\subset{\mathbb R^n}$, $n \geq 3$, we have \[ f_K(\theta)=C(vol_{n-1}(K\cap \theta^{\perp}))^{n+1}\qquad\forall \theta\in S^{n-1}, \] where the constant $C$ is independent of $\theta$, must $K$ be an ellipsoid Here, $f_K$ is the is the curvature function (the reciprocal of the Gaussian curvature). We will show that the answer is affirmative for $K$ close enough to the Euclidean ball in the Banach-Mazur distance.