The Generalized Busemann-petty Problem with Weights
نویسنده
چکیده
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 < i ≤ n − 1, the negative answer was given by Bourgain and Zhang [1]; see also [8], [12]. For the special case, when K is a body of revolution, the answer for i = 2 and 3 is affirmative [5], [14], [12]. The case, when K is an arbitrary origin-symmetric convex body and i = 2 and 3, is still open. In a recent paper [16], Zvavitch considered the Busemann-Petty problem (i = n − 1) in a more general setting, when volumes under consideration are evaluated with respect to general measures satisfying
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