A note on 𝑘-intersection bodies

Schlieper, Jared

doi:10.1090/S0002-9939-07-08774-6

A note on $k$-intersection bodies
HTML articles powered by AMS MathViewer

by Jared Schlieper PDF

Proc. Amer. Math. Soc. 135 (2007), 2081-2088 Request permission

Abstract:

The concept of an intersection body is central for the dual Brunn-Minkowski theory and has also played an important role in the solution of the Busemann-Petty problem. A more general concept of $k$-intersection bodies is related to the generalization of the Busemann-Petty problem. In this note, we compare classes of $k$-intersection bodies for different $k$ and examine the conjecture that these classes increase with $k$. In particular, we construct a $4$-intersection body that is not a $2$-intersection body.

References

I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New York-London, 1964. Translated by Eugene Saletan. MR 0166596
I. M. Gel’fand and N. Ya. Vilenkin, Generalized functions. Vol. 4: Applications of harmonic analysis, Academic Press, New York-London, 1964. Translated by Amiel Feinstein. MR 0173945
N. J. Kalton and A. Koldobsky, Banach spaces embedding isometrically into $L_p$ when $0<p<1$, Proc. Amer. Math. Soc. 132 (2004), no. 1, 67–76. MR 2021249, DOI 10.1090/S0002-9939-03-07169-7
Alexander Koldobsky, A Banach subspace of $L_{1/2}$ which does not embed in $L_1$ (isometric version), Proc. Amer. Math. Soc. 124 (1996), no. 1, 155–160. MR 1285999, DOI 10.1090/S0002-9939-96-03010-9
Alexander Koldobsky, Positive definite distributions and subspaces of $L_{-p}$ with applications to stable processes, Canad. Math. Bull. 42 (1999), no. 3, 344–353. MR 1703694, DOI 10.4153/CMB-1999-040-5
Alexander Koldobsky, Fourier analysis in convex geometry, Mathematical Surveys and Monographs, vol. 116, American Mathematical Society, Providence, RI, 2005. MR 2132704, DOI 10.1090/surv/116
A. Koldobsky, A generalization of the Busemann-Petty problem on sections of convex bodies, Israel J. Math. 110 (1999), 75–91. MR 1750442, DOI 10.1007/BF02808176
Erwin Lutwak, Intersection bodies and dual mixed volumes, Adv. in Math. 71 (1988), no. 2, 232–261. MR 963487, DOI 10.1016/0001-8708(88)90077-1
E. Milman, Generalized intersection bodies, preprint.
Rolf Schneider, Zonoids whose polars are zonoids, Proc. Amer. Math. Soc. 50 (1975), 365–368. MR 470857, DOI 10.1090/S0002-9939-1975-0470857-2