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Intersection bodies and the Busemann-Petty inequalities in \(\mathbb{R}^ 4\). (English) Zbl 0826.52011

The following question is known as the Busemann-Petty problem. Let \(K_1\), \(K_2\) be convex bodies in Euclidean space \(\mathbb{R}^n\), centrally symmetric with respect to the origin. Let \(\lambda_k\) denote \(k\)- dimensional volume. If \[ \lambda_{n - 1} (K_1 \cap H) \leq \lambda_{n - 1} (K_2 \cap H) \] for all hyperplanes \(H\) through the origin, does it follow that \[ \lambda_n (K_1) \leq \lambda_n (K_2)? \] The present paper extends the negative answer, known before for \(n \geq 7\), to \(n \geq 4\). This is optimal, since R. J. Gardner [Ann. Math., II. Ser. 140, No. 2, 435–437 (1994; Zbl 0826.52010)] gave a positive answer for \(n = 3\). The proof is based on the equivalence of a negative answer to the Busemann-Petty problem and the existence of centrally symmetric non-intersection bodies, as established earlier by R. J. Gardner [Trans. Am. Math. Soc. 342, No. 1, 435-445 (1994; Zbl 0801.52005)] and the author [Trans. Am. Math. Soc. 345, No. 2, 777–801 (1994; Zbl 0812.52005)]. The crucial step in this paper is to show that no cube in \(\mathbb{R}^n\) \((n \geq 4)\) is an intersection body. It is also proved that no cylinder in \(\mathbb{R}^n\) \((n \geq 5)\) is an intersection body, and that \(C^\infty\) symmetric bodies of revolution in \(\mathbb{R}^3\) and \(\mathbb{R}^4\) are intersection bodies.
Editorial remark: Later, the author corrected this publication towards a positive answer for \(n=4\), see [G. Zhang, Ann. Math. (2) 149, No. 2, 535–543 (1999; Zbl 0937.52004)].

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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