Chebyshev potentials, Fubini--Study metrics, and geometry of the space of Kähler metrics

The Chebyshev potential of a Kähler potential on a projective variety, introduced by Witt Nyström, is a convex function defined on the Okounkov body. It is a generalization of the symplectic potential of a torus-invariant Kähler potential on a toric variety, introduced by Guillemin, that is a convex function on the Delzant polytope. A folklore conjecture asserts that a curve of Chebyshev potentials associated to a curve in the space of Kähler potentials is linear in the time variable if and only if the latter curve is a geodesic in the Mabuchi metric. This is classically true in the special toric setting, and in general Witt Nyström established the sufficiency. The goal of this article is to disprove this conjecture. More generally, we characterize the Fubini--Study geodesics for which the conjecture is true on projective space. The proof involves explicitly solving the Monge--Ampère equation describing geodesics on the subspace of Fubini--Study metrics and computing their Chebyshev potentials.