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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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