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The Two-Piece Property and Tight n-Manifolds-with-Boundary in E^{n}

by Thomas F. Banchoff

A set A in E^{n} is said to have the two-piece property if every hyperplane
in E^{n} cuts A into at most two pieces. The TPP is a generalized notion
of convexity and it reduces to minimum total absolute curvature
when A is a compact 2-manifold. In this paper, we prove that a connected
compact 2-manifold-with-boundary in E^{2} has the TPP if and only
if every component of the boundary has the TPP. This result
does not generalize to higher dimensions unless additional assumptions
are made. The concept of *k-tightness* is introduced and
we prove that an (n+1)-manifold-with-boundary, M, embedded in
E^{(n+1)} is 0- and (n-1)-tight if and only if its boundary is also
0- and (n-1)-tight.