A sharp quantitative version of Hales' isoperimetric honeycomb theorem
  We prove a sharp quantitative version of Hales' isoperimetric honeycomb theorem by exploiting a quantitative isoperimetric inequality for polygons and an improved convergence theorem for planar bubble clusters. Further applications include the description of isoperimetric tilings of the torus with respect to almost unit-area constraints or with respect to almost flat Riemannian metrics. 
Marco Caroccia and Francesco Maggi
math.AP math.MG math.OC