On a Generalization of the Busemann–Petty Problem
نویسندگان
چکیده
The generalized Busemann–Petty problem asks: If K and L are origin-symmetric convex bodies in Rn, and the volume of K ∩H is smaller than the volume of L ∩H for every i-dimensional subspace H, 1 < i < n, does it follow that the volume of K is smaller than the volume of L? The hyperplane case i = n−1 is known as the Busemann–Petty problem. It has a negative answer when n > 4, and has a positive answer when n = 3, 4. This paper gives a negative answer to the generalized Busemann–Petty problem for 3 < i < n in the stronger sense that the integer i is not fixed. For the 2-dimensional case i = 2, it is proved that the problem has a positive answer when L is a ball and K is close to L.
منابع مشابه
The Australian Journal of Mathematical Analysis and Applications
The norm defined by Busemann’s inequality establishes a class of star body intersection body. This class of star body plays a key role in the solution of Busemann-Petty problem. In 2003, Giannapoulos [1] defined a norm for a new class of half-section. Based on this norm, we give a geometric generalization of Busemann-Petty problem, and get its answer as a result.
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