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- W2069985237 abstract "Abstract Determining meaningful lower bounds on the supremal strict p -negative type of classes of finite metric spaces is a difficult nonlinear problem. In this paper we use an elementary approach to obtain the following result: given a finite metric space ( X , d ) there is a constant ζ >0, dependent only on n =∣ X ∣ and the scaled diameter 𝔇=( diamX )/min{ d ( x , y )∣ x ⁄= y } of X (which we may assume is >1), such that ( X , d ) has p -negative type for all p ∈[0, ζ ] and strict p -negative type for all p ∈[0, ζ ). In fact, we obtain A consideration of basic examples shows that our value of ζ is optimal provided that 𝔇≤2. In other words, for each 𝔇∈(1,2] and natural number n ≥3, there exists an n -point metric space of scaled diameter 𝔇 whose supremal strict p -negative type is exactly ζ . The results of this paper hold more generally for all finite semi-metric spaces since the triangle inequality is not used in any of the proofs. Moreover, ζ is always optimal in the case of finite semi-metric spaces." @default.
- W2069985237 created "2016-06-24" @default.
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- W2069985237 date "2009-08-19" @default.
- W2069985237 modified "2023-10-14" @default.
- W2069985237 title "OPTIMAL LOWER BOUND ON THE SUPREMAL STRICT <i>p</i>-NEGATIVE TYPE OF A FINITE METRIC SPACE" @default.
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- W2069985237 doi "https://doi.org/10.1017/s0004972709000604" @default.
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