TheMorse Theory of Čech and Delaunay Complexes
نویسنده
چکیده
Consider a sphere S enclosing the points Q ⊆ X and excluding the points E ⊆ X • Let encl S be all points of X enclosed by S, and excl S excluded by S • Let on S be the points of X on S • Write the center of S as an a?ne combination zS = ∑x∈on S μxx • Let front S = {x ∈ on S ∣ μx > 0}, and back S = {x ∈ on S ∣ μx < 0} The Karush–Kuhn–Tucker conditions for the sphere minimization problem yield: Lemma A sphere S enclosingQ and excluding E is the smallest such sphere i>: • zS ∈ a (on S) • Q ∈ [front S , encl S] (i.e., front S ⊆ Q ⊆ encl S) • E ∈ [back S , excl S] The Čech intervals are the level sets f −1 C (t) of the Čech function:
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