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Geometric Aspects of Functional Analysis pp 413–422Cite as

On Isotropicity with Respect to a Measure

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2116))

Abstract

A body C is said to be isotropic with respect to a measure μ if the function

$$\displaystyle{\theta \rightarrow \int _{C}\left \langle x,\theta \right \rangle ^{2}d\mu (x)}$$

is constant on the unit sphere. In this note, we extend a result of Bobkov, and prove that every body can be put in isotropic position with respect to any rotation invariant measure.When the body C is convex, and the measure μ is log-concave, we relate the isotropic position with respect to μ to the famous M-position, and give bounds on the isotropic constant.

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Acknowledgements

I would like to thank Prof. Sergey Bobkov for fruitful and interesting discussions, and my advisor, Prof. Vitali Milman, for his help and support. I would also like to thank the referee for his useful remarks and corrections. The author is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. The author is also supported by ISF grant 826/13 and BSF grant 2012111.

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Authors and Affiliations

  1. School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Liran Rotem

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  1. Liran Rotem
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Correspondence to Liran Rotem .

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Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Bo'az Klartag

  2. Technion - Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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Rotem, L. (2014). On Isotropicity with Respect to a Measure. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_27

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