What is the minimal possible value of the maximal total sidelength shared by any two tiles in a tiling of the plane if all tiles have the same area A?
total sidelength = Length-integral of the curve formed by the intersection of two tiles
i) Using a finite set of tiles
ii) Using any set of tiles.
This is known for 3D as http://mathworld.wolfram.com/KelvinsConjecture.html
I have found no info on the planar case, so it may be trivial, in which case I want to see the proof.