A note on Seshadri constants, Gromov widths of toric surfaces and lattice widths of polygons

In this note, we study Seshadri constants and Gromov widths of toric surfaces via lattice widths of their moment polygons. We give the sharp lower bound of the ratio between the Gromov width of a symplectic toric $4$-fold and the lattice width of the moment polygon, which answers to a question raised by Codenotti, Hall and Hofscheier.