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1971 (Publication #13)

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

Trans. Amer. Math. Soc. 161 (1971), 259-267.

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ABSTRACT

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.