Second Derivative Test for Intersection Bodies
In 1956, Busemann and Petty asked whether symmetric convex bodies in ℝn with larger central hyperplane sections alsohave greater volume. This question was answered in the negativefor n>=5 in a series of papers giving individual counterexamples.In 1988, Lutwak introduced the concept of an intersection bodyand proved that every smooth nonintersection body in ℝn providesa counterexample to the Busemann-Petty problem. In thisarticle, we use the connection between intersection bodies andpositive definite distributions, established by the author inan earlier paper, to give a necessary condition for intersectionbodies in terms of the second derivative of the norm. This resultallows us to produce a variety of counterexamples to the Busemann-Pettyproblem in ℝn, n>=5. For example, the unit ball of theq-sum of any finite dimensional normed spaces X and Y with q>2,dim(X)>=1, dim(Y)>=4 is not an intersection body, aswell as the unit balls of the Orlicz spaces ℓnM , n>=5,with M\'(0)=M\"(0)=0. Copyright 1998 Academic Press.
Document Type: Research Article
Division of Mathematics and Statistics, University of Texas at San Antonio, San Antonio, Texas, 78249
Publication date: June 1, 1998
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