On Isoperimetric Inequalities for Log-convex Measures

Let μ = ρdx be a Borel measure on Rd. A Borel set A ⊂ R is a solution of the isoperimetric problem if for any B ⊂ R satisfying μ(A) = μ(B) one has μ(∂A) ≤ μ(∂B), where μ(∂A) = ∫ ∂A ρ dHd−1 is the corresponding surface measure. There exists only a small number of examples where the isoperimetric problem has an exact solution. The most important case is given by Lebesgue measure λ on R, the solutions for the isoperimetric problem are the balls. Recall the corresponding isoperimetric inequality