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All principal orbits of the standard Hamiltonian $T^n$-action on the complexprojective space ${\mathbb C}P^n$ are Lagrangian tori.In this article, we provethat most of them are not volume minimizing under Hamiltonian isotopies of${\mathbb C}P^n$ if the complex dimension $n$ is greater than two, althoughthey are Hamiltonian minimal and Hamiltonian stable.

Almost all Lagrangian torus orbits in ${\mathbb C}P^n$ are not Hamiltonian volume minimizing