Relating transfinite diameters using an Okounkov body

We derive relations between the transfinite diameter of a locally circled subset K of the complexified sphere in $$\mathbb {C}^3$$ and notions of weighted transfinite diameter of the projection of K to $$\mathbb {C}^2$$ . Our method is based on connecting a Chebyshev transform of K on an Okounkov body of V to the classical Chebyshev transform of the projection of K to $$\mathbb {C}^2$$ .