The two-piece property is a useful generalization of convexity for subsets of n-dimensional Euclidean space. Simply put, a subset of Eⁿ has the two-piece property (TPP) if any hyperplane cuts it into at most two connected pieces. It is not too difficult to see that convex sets will have the TPP. A more restrictive variant on the TPP is the spherical two-piece property (STPP), which in two dimensions is known as the circular two-piece property (CTPP). A subset of Eⁿ has the STPP if every (n-1)-sphere or hyperplane cuts it into at most two pieces.

In this paper we examine a third variant of the TPP, by considering the spheres of a different metric, namely the maximum-value metric of Rⁿ. That is, distances in Rⁿ are measured by the maximum difference in coordinates. The resulting property, analogous to the STPP, is known as the grid-cube two-piece property (GTPP). The spheres in the maximum-value metric are the grid-hypercubes of Rⁿ, the set of points a fixed distance (in this metric) from a fixed center. Thus, a subset of Rⁿ has the GTPP is every grid hypercube cuts it into at most two pieces.

This paper begins with the case of the GTPP and curves in the plane.