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Example text
Yet, in convex domains the constant C2 in (1) is geometry independent [4]. Defining the distance defect ratio of a pair of points x, y ∈ cl(Ω) = Ω ∪ ∂Ω (with ∂Ω the boundary of Ω) by µ(x, y)Ω = ρ(x, y)Ω |x − y| (2) where ρ(x, y)Ω is the length of the shortest path inside cl(Ω) connecting x and y, we observe that in the domains {Ωǫ } of the example, there exist pairs of points with distance defect ratio growing as ǫ → 0. Note that there is no upper bound for the distance defect ratio of arbitrary domains, while in convex domains the distance defect ratio is 1.
Both the similarity and the dissimilarity should be examinable in a clear and meaningful way. Here, we give the simple mathematical descriptions of partitional clustering and hierarchical clustering, based on [40]. Given a set of N input patterns X = {x1 , . . , xj , . . , xN }, where xj = (xj1 , xj2 , . . , xjd )T ∈ Rd and each xji measure is said to be a feature (attribute, dimension, or variable), 32 R. Xu, D. Wunsch II • (Hard) partitional clustering attempts to seek a K-partition of X, C = {C1 , .
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