Second Derivative Test for Intersection Bodies

✍ Scribed by Alexander Koldobsky

• Book ID: 102562479
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 224 KB
• Volume: 136
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1998.1719

✦ Synopsis

In 1956, Busemann and Petty asked whether symmetric convex bodies in R n with larger central hyperplane sections also have greater volume. This question was answered in the negative for n 5 in a series of papers giving individual counterexamples. In 1988, Lutwak introduced the concept of an intersection body and proved that every smooth nonintersection body in R n provides a counterexample to the Busemann Petty problem. In this article, we use the connection between intersection bodies and positive definite distributions, established by the author in an earlier paper, to give a necessary condition for intersection bodies in terms of the second derivative of the norm. This result allows us to produce a variety of counterexamples to the Busemann Petty problem in R n , n 5. For example, the unit ball of the q-sum of any finite dimensional normed spaces X and Y with q>2, dim(X) 1, dim(Y) 4 is not an intersection body, as well as the unit balls of the Orlicz spaces l n M , n 5, with M$(0)=M"(0)=0.