A body C is said to be isotropic with respect to a measure µ if the function (Formula presented.) 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.
Bibliographical noteFunding Information:
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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