A positive solution to the Busemann-Petty problem in \(\mathbb{R}^4\)
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Publication:1293368
DOI10.2307/120974zbMATH Open0937.52004arXivmath/9903205OpenAlexW2172026658MaRDI QIDQ1293368
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Publication date: 25 May 2000
Published in: (Search for Journal in Brave)
Abstract: H. Busemann and C. M. Petty posed the following problem in 1956: If K and L are origin-symmetric convex bodies in R^n and for each hyperplane H through the origin the volumes of their central slices satisfy vol(K cap H) < vol(L cap H), does it follow that the volumes of the bodies themselves satisfy vol(K) < vol(L)? The problem is trivially positive in R^2. However, a surprising negative answer for n <= 12 was given by Larman and Rogers in 1975. Subsequently, a series of contributions were made to reduce the dimensions to n >= 5 by a number of authors. That is, the problem has a negative answer for n >= 5. It was proved by Gardner that the problem has a positive answer for n=3. The case of n=4 was considered in [Ann. of Math. (2) 140 (1994), 331-346], but the answer given there is not correct. This paper presents the correct solution, namely, the Busemann-Petty problem has a positive solution in R^4, which, together with results of other cases, brings the Busemann-Petty problem to a conclusion.
Full work available at URL: https://arxiv.org/abs/math/9903205
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