As we have mentioned before, the regular TPP is a generalization of convexity. In [2], Banchoff lays out the foundations of the theory in two dimensions, which we briefly repeat here, and continues to prove the higher-dimensional result using k-tightness. We begin by defining the TPP in two dimensions:

Definition. A path connected set A has the two-piece property (TPP) if and only if the intersection of H and A is path connected for any closed half-plane H in E².

Remark. Any convex set A has the TPP since the intersection of H and A will be convex for any H, and therefore path connected.

The following propositions characterize all curves in the plane with the TPP. They are presented here without proof (which can be found in [2]).

Proposition. If A is a closed curve which is the boundary of a bounded region R in E², then A has the TPP if and only if R is a convex region.

Proposition. If A is a nonclosed curve in E², with or without endpoints, then A has the TPP if and only if A is a connected subset of a line.

The next proposition demonstates another example of a set with the TPP: a convex set with a finite number of disjoint convex sets removed, as in the illustration below. The theorem following shows that all convex sets with the TPP look like those described in the proposition in an essential way.

Proposition. If A is a closed convex set and if {B_i}, i = 1, 2, ..., m, is a finite collection of bounded open convex sets with disjoint boundaries (and interiors disjoint from the boundary of A), then A \ {union B_i} has the TPP.

Theorem. If A is a closed set in E² with the TPP, then each of the bounded components B_i of E² - A is a convex set and the union of A and the bounded components of its complement is a convex set.

In order to reasonably restrict the class of sets we're looking at, and at the same time make the transition to higher dimensions easier, it is convenient to restate the above results in terms of manifolds.

Definition. A 2-manifold-with-boundary M² embedded in E² is a closed point set M² which can be expressed as a union of two sets: M²º, the interior of M² = {p in M² such that there is a disc neighborhood B² of p in E² with B² contained in M²} and the boundary of M² = {q in M² such that there is a neighborhood B² of q in E² with the intersection of B² with M² given by the 1-1 continuous image of the intersection of an open disc about the origin with the closed upper half-plane, with q corresponding to the origin}.

The analogous result of the previous theorem for 2-manifolds-with-boundary follows:

Theorem. A 2-manifold-with-boundary M² embedded in E² has the TPP if and only if each bounded component of the boundary of M² has the TPP and if the union of M² with the bounded component of its complement is a convex set.

For a similar treatment of the circle two-piece property, go here. Otherwise you can continue with the GTPP of curves in the plane.