Quantitative stability in the isodiametric inequality via the   isoperimetric inequality
  The isodiametric inequality is derived from the isoperimetric inequality trough a variational principle, establishing that balls maximize the perimeter among convex sets with fixed diameter. This principle brings also quantitative improvements to the isodiametric inequality, shown to be sharp by explicit nearly optimal sets. 
Francesco Maggi, Marcello Ponsiglione and Aldo Pratelli
math.MG