On the Classification of Origin-symmetric Convex Lattice Sets

This paper studies the classification of origin-symmetric convex lattice sets under integer unimodular transformations. For an origin-symmetric convex lattice set, we construct an affine invariant and consider the affine invariance of convex lattice sets from different points of view. Furthermore, we provide an upper bound of coordinates of points in origin-symmetric convex lattice sets, which is accomplished by proving the upper bound of the minimum area of the square enclosing an origin-symmetric convex lattice set. Using these results we classify origin-symmetric convex lattice sets with cardinality not greater than 17.