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The isomorphic Busemann–Petty problem for s-concave measures

  • Denghui WuEmail author
Original Paper
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Abstract

This paper considers the isomorphic lower-dimensional Busemann–Petty problem. Fix \(k\in \{1,2,\ldots ,n-1\}\). If \(\mu \) is a measure on \(\mathbb {R}^n\) with an even non-negative density, and KL are origin-symmetric convex bodies in \(\mathbb {R}^n\) such that for every \((n-k)\)-dimensional subspaces H of \(\mathbb {R}^n\),
$$\begin{aligned}\mu (K\cap H)\le \mu (L\cap H),\end{aligned}$$
does there exist a constant \(\mathcal {L}\) such that
$$\begin{aligned}\mu (K)\le {\mathcal {L}}^\frac{nk}{n-k}\mu (L)?\end{aligned}$$
We provide different new estimates for the constant \(\mathcal {L}\) for arbitrary s-concave measures, containing the cases \(0<s<1/n\), \(s<0\) and \(\log \)-concave cases.

Keywords

Convex bodies Busemann–Petty problem Measures Intersection bodies 

Mathematics Subject Classification (2010)

52A20 

Notes

Acknowledgements

I would like to thank University of Missouri-Columbia for the hospitality during my stay there. I am grateful to Professor Alexander Koldobsky for drawing my attention to the subject of this article and stimulating discussions. I also would like to thank Professor Artem Zvavitch for reading the original manuscript, and anonymous referees for helpful comments that directly lead to the improvement of the original manuscript. This work was supported, in partial, by the Initial Foundation for Scientific Research of Northwest A&F University (2452018016, 2452018074) and China Scholarship Council (201606990052).

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of ScienceNorthwest A&F UniversityYanglingChina

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