## 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)

### Citations:

Zbl 0826.52010; Zbl 0801.52005; Zbl 0812.52005; Zbl 0937.52004
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