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We prove a sharp quantitative version of Hales' isoperimetric honeycombtheorem by exploiting a quantitative isoperimetric inequality for polygons andan improved convergence theorem for planar bubble clusters. Furtherapplications include the description of isoperimetric tilings of the torus withrespect to almost unit-area constraints or with respect to almost flatRiemannian metrics.

A sharp quantitative version of Hales' isoperimetric honeycomb theorem