The two-piece property and tight $n$-manifolds-with-boundary in $E^{n}$

Abstract:

The two-piece property for a set A is a generalization of convexity which reduces to the condition of minimal total absolute curvature if A is a compact 2-manifold. We show that a connected compact 2-manifold-with-boundary in ${E^2}$ has the TPP if and only if each component of the boundary has the TPP. The analogue of this result is not true in higher dimensions without additional conditions, and we introduce a stronger notion called k-tightness and show that an $(n + 1)$-manifold-with-boundary ${M^{n + 1}}$ embedded in ${E^{n + 1}}$ is 0- and $(n - 1)$-tight if and only if its boundary is 0- and $(n - 1)$-tight.