For a metric measure space and a probability measure μ we can define the Minkowski content of a measurable set as: μ+(A) = lim inf s→0+ μ(As) − μ(A)/s. For such a space we can define the isoperimetric profile in accordance with the classical isoperimetric profile through: Iμ(t) = inf{μ+(A) : ABorel, μ(A) = t} An equivalent formulation of the so called Kannan - Lovasz - Simmonovitz conjecture, is that the isoperimetric profile is bounded from below by a constant independent of the dimension for log-concave measures in Rd . In this talk, we will present shortly the main results towards the proof of this conjecture, and we will prove that the concture holds up to a polylog.