Second Derivative Test for Intersection Bodies

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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

Affiliations: Division of Mathematics and Statistics, University of Texas at San Antonio, San Antonio, Texas, 78249

Publication date: June 1, 1998

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