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Geometric methods of convex polytopes are applied to demonstrate a
new connection between convexity and threshold logic. A
cut-complex is a cubical complex whose vertices are strictly
separable from rest of the vertices of the $n$-cube by a
hyperplane of $R^n$. Cut-complexes are geometric presentations for
threshold Boolean functions and thus are thus are related to
threshold logic. For an old classical theorem of threshold logic a
shorter but geometric proof is given. The dimension of the cube
hull of a cut-complex is shown to be the same as the maximum
degree of the vertices in the complex. A consequence of the latter
result indicates that any two isomorphic cut-complexes are
isometric.
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