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Geometrical Bell Inequalities (GBIs) are the strongest known Bellinequalities for collections of qubits. However, their generalizations to othersystems is not yet fully understood. We formulate GBIs for an arbitrary number$N$ of observers, each of which possesses a particle of an arbitrary dimension$d$. The whole $(d-1)$-parameter family of local observables with eigenbasesunbiased to the computational basis is used, but it is immediate to use adiscrete subset of them. We argue analytically for qutrits and numerically forother systems that the violations grows exponenetially with $N$. Within thestudied range, the violation also grows with $d$. Interestingly, we observethat the logarithm of the violation ratio for ququats grows with $N$ slowerthan the doubled logarithm of the violation ratio for qubits, which implies akind of subadditivity.

Subadditivity of logarithm of violation of geometric Bell inequalities for qudits