Uniqueness of a smooth convex body with uniform cone volume measure in the neighborhood of a ball

We prove that in every n-dimensional there exists a constant c=c(n)>0 so that in the c(n)-neighborhood of a ball, the only convex body with uniform cone volume measure is the ball. The goal of the talk will be to give an insight into some analytic aspects of the Log-Brunn-Minkowski and Log-Minkowski conjectures made by Boroczky, Lutwak, Yang and Zhang. This talk is based on the joint papers with Colesanti, Marsiglietti and Colesanti.

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