A converse to Maz'ya's inequality for capacities under curvature lower bound

We survey some classical inequalities due to Maz'ya relating isocapacitary inequalities with their functional and isoperimetric counterparts in a measure-metric space setting, and extend Maz'ya's lower bound for the $q$-capacity ($q>1$) in terms of the 1-capacity (or isoperimetric) profile. We then proceed to describe results by Buser, Bakry, Ledoux and most recently by the author, which show that under suitable convexity assumptions on the measure-metric space, Maz'ya's inequality for capacities may be reversed, up to dimension independent numerical constants: a matching lower bound on 1-capacity may be derived in terms of the $q$-capacity profile. We extend these results to handle arbitrary $q > 1$ and weak semi-convexity assumptions, by obtaining some new delicate semi-group estimates.