Any gap-free packing of a bag by countably many positive-volume sets that touch on zero-area sets must have infinite total surface area.
Ambrosio, Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv
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On the total surface area of potato packings
Any gap-free packing of a bag by countably many positive-volume sets that touch on zero-area sets must have infinite total surface area.