On Form Factors in N=4 SYM Theory and Polytopes

In this paper we discuss different recursion relations (BCFW and all-line shift) for the form factors of the operators from the $\mathcal{N}=4$ SYM stress-tensor current supermultiplet $T^{AB}$ in momentum twistor space. We show that cancelations of spurious poles and equivalence between different types of recursion relations can be naturally understood using geometrical interpretation of the form factors as special limit of the volumes of polytopes in $\mathbb{C}\mathbb{P}^4$ in close analogy with the amplitude case. We also show how different relations for the IR pole coefficients can be easily derived using momentum twistor representation. This opens an intriguing question - which of powerful on-shell methods and ideas can survive off-shell ?