Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitative versions of these inequalities involving the so-called
p -negative type gap. In particular, we focus our attention on the class of finite ultrametric spaces which are important in areas such as phylogenetics and data mining. Let
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α. Then the
p -negative type gap
856&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=a89a72507b75dac00e2e758d997e59ea" title="Click to view the MathML source">ΓX(p) of
856&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=2c619669fd7d5cb26a068936bb1f487c" title="Click to view the MathML source">(X,d) is positive for all
856&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=f930879dd72285404df7e00e42badcc0" title="Click to view the MathML source">p≥0. In this paper we compute the value of the limit
It turns out that this value is positive and it may be given explicitly by an elegant combinatorial formula. This formula allows us to characterize when the ratio
856&_mathId=si5.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=e4808912dc368c5cfb5c361ec1205a75" title="Click to view the MathML source">ΓX(p)/αp is a constant independent of
p . The determination of
856&_mathId=si321.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=0e9ef2006b2509bac73a4ca6738f064e" title="Click to view the MathML source">ΓX(∞) also leads to new, asymptotically sharp, families of enhanced
p -negative type inequalities for
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p, the inequality
holds for each finite subset
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n -tuple
856&_mathId=si10.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=7f1eb2b2fdb15313d6129848871ba4b5" title="Click to view the MathML source">ζ=(ζ1,…,ζn)∈Rn that satisfies
856&_mathId=si11.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=1c195f8e4e472e777093e796833384d2" title="Click to view the MathML source">ζ1+⋯+ζn=0. Notably, these results do not extend to general finite metric spaces.