Concentration in a thin Euclidean shell for log-concave measures

A weak version of a conjecture stated by Kannan, Lovász and Simonovits claims that an isotropic log-concave probability μ   on RnRn should be concentrated in a thin Euclidean shell in the following way:equation(1)∀t∈[0,nκ],μ{x∈Rn:(1−tnκ)⩽|x|n⩽(1+tnκ)}⩾1−Ce−ct where κ=1/2κ=1/2 and c and C   are positive absolute constants. For κ=1/10.02κ=1/10.02, this inequality has been established by Klartag. By combining different approaches introduced by Klartag and by Guédon, Paouris and the author, we improve this result by showing that the inequality (1) holds with κ=1/8κ=1/8.