I will consider periodic tessellations of the Euclidean space arising from the minimization of local or non-local perimeter functionals, discussing existence, regularity and qualitative properties of minimizers. I will mainly focus on the case of nonlocal perimeter functionals and discuss some related open problem.
Based on joint works with Ilaria Fragalà (Politecnico di Milano) and Matteo Novaga (Università di Pisa).