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A positive answer to the Busemann-Petty problem in three dimensions. (English) Zbl 0826.52010

In this paper, the author provides the keystone to the complete solution of the long-standing Busemann-Petty problem, by proving the following result. If \(K_1, K_2 \subset \mathbb{R}^3\) are convex bodies, centrally symmetric with respect to the origin and satisfying \[ \lambda_2 (K_1 \cap H) \leq \lambda_2 (K_2 \cap H) \] for all planes \(H\) through \(0\), then \[ \lambda_3 (K_1) \leq \lambda_3(K_2); \] here \(\lambda_k\) denotes \(k\)-dimensional volume. Essential tools of the proof are Lutwak’s intersection bodies, as in earlier contributions to the problem, and a new inversion formula for a spherical Radon transform which is the radial function of a centered convex body.

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
52A38 Length, area, volume and convex sets (aspects of convex geometry)
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