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Alexander Koldobsky - A functional analytic approach to the Busemann-Petty problem on sections of convex bodies
ALEXANDER KOLDOBSKY, University of Texas at San Antonio, San Antonio, Texas 78249, USA | |
A functional analytic approach to the Busemann-Petty problem on sections of convex bodies |
The 1956 Busemann-Petty problem asks whether symmetric convex bodies in with larger central hyperplane sections must also have greater volume. The solution to the problem has recently been completed, and the answer is negative if and affirmative when . We show a more general result, where the inequalities for the volume of central sections are replaced by similar inequalities for the derivatives of the parallel section functions. For example, if n is an even integer, K and L are origin-symmetric convex bodies in with C(n-4)-boundaries and the (n-4)-th derivatives of the parallel section functions of K at zero (in every direction) are smaller than the corresponding derivatives for L, then . However, if n is odd, similar inequalities for the (n-5)-th derivatives do not necessarily imply that the volume of K is smaller.
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