Intersection bodies and the Busemann-Petty problem
نویسندگان
چکیده
منابع مشابه
The Complex Busemann-petty Problem on Sections of Convex Bodies
The complex Busemann-Petty problem asks whether origin symmetric convex bodies in C with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if n ≤ 3 and negative if n ≥ 4.
متن کاملBusemann - Petty problem on sections of convex bodies
The Busemann-Petty problem asks whether origin-symmetric convex bodies in R n with smaller central hyperplane sections necessarily have smaller n-dimensional volume. It is known that the answer is affirmative if n ≤ 4 and negative if n ≥ 5. In this article we modify the assumptions of the original Busemann-Petty problem to guarantee the affirmative answer in all dimensions.
متن کاملThe Generalized Busemann-petty Problem with Weights
This question is known as the generalized Busemann-Petty problem. For i = n − 1, the problem was posed by Busemann and Petty [2] in 1956. It has a long history, and the answer is affirmative if and only if n ≤ 4; see [3], [8], [11]. For the generalized Busemann-Petty problem the following statements are known. If i = 2, n = 4, an affirmative answer follows from that in the case i = n − 1. If 3 ...
متن کاملThe Lower Dimensional Busemann-petty Problem for Bodies with the Generalized Axial Symmetry
The lower dimensional Busemann-Petty problem asks, whether n-dimensional origin-symmetric convex bodies, having smaller i-dimensional sections, necessarily have smaller volumes. For i = 1, the affirmative answer is obvious. For i > 3, the answer is negative. For i = 2 and i = 3, the problem is still open, except when the body with smaller sections is a body of revolution. In this case the answe...
متن کاملThe Busemann-petty Problem in Hyperbolic and Spherical Spaces
The Busemann-Petty problem asks whether origin-symmetric convex bodies in R with smaller central hyperplane sections necessarily have smaller n-dimensional volume. It is known that the answer to this problem is affirmative if n ≤ 4 and negative if n ≥ 5. We study this problem in hyperbolic and spherical spaces.
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1994
ISSN: 0002-9947,1088-6850
DOI: 10.1090/s0002-9947-1994-1201126-7