Negative type inequalities arise in the study of embedding properties of metric sp
aces, but they often reduce to intr
actable 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 sp
aces which are important in areas such as phylogenetics and data mining. Let
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ace with minimum non-zero distance
α. Then the
p -negative type gap
05856&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=a89a72507b75dac00e2e758d997e59ea" title="Click to view the MathML source">ΓX(p) of
05856&_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
05856&_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 char
acterize when the ratio
05856&_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
05856&_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
05856&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=2c619669fd7d5cb26a068936bb1f487c" title="Click to view the MathML source">(X,d). Indeed, suppose that
05856&_mathId=si66.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=35b8bc05e92e30d66e0be2cbe3d617fc" title="Click to view the MathML source">G∈(0,ΓX(∞)). Then, for all sufficiently large
p, the inequality
holds for e
ach finite subset
05856&_mathId=si68.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=324e2c53a3652686fa30fcc6056a8f21" title="Click to view the MathML source">{z1,…,zn}⊆X, and e
ach scalar
n -tuple
05856&_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
05856&_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 sp
aces.