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
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=2c619669fd7d5cb26a068936bb1f487c" title="Click to view the MathML source">(X,d)class="mathContainer hidden">class="mathCode"> be a given finite ultrametric sp
ace with minimum non-zero distance
α. Then the
p -negative type gap
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=a89a72507b75dac00e2e758d997e59ea" title="Click to view the MathML source">ΓX(p)class="mathContainer hidden">class="mathCode"> of
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=2c619669fd7d5cb26a068936bb1f487c" title="Click to view the MathML source">(X,d)class="mathContainer hidden">class="mathCode"> is positive for all
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=f930879dd72285404df7e00e42badcc0" title="Click to view the MathML source">p≥0class="mathContainer hidden">class="mathCode">. In this paper we compute the value of the limit
class="formula" id="fm0010">
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
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si5.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=e4808912dc368c5cfb5c361ec1205a75" title="Click to view the MathML source">ΓX(p)/αpclass="mathContainer hidden">class="mathCode"> is a constant independent of
p . The determination of
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si321.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=0e9ef2006b2509bac73a4ca6738f064e" title="Click to view the MathML source">ΓX(∞)class="mathContainer hidden">class="mathCode"> also leads to new, asymptotically sharp, families of enhanced
p -negative type inequalities for
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=2c619669fd7d5cb26a068936bb1f487c" title="Click to view the MathML source">(X,d)class="mathContainer hidden">class="mathCode">. Indeed, suppose that
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si66.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=35b8bc05e92e30d66e0be2cbe3d617fc" title="Click to view the MathML source">G∈(0,ΓX(∞))class="mathContainer hidden">class="mathCode">. Then, for all sufficiently large
p, the inequality
class="formula" id="fm0020">
holds for e
ach finite subset
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si68.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=324e2c53a3652686fa30fcc6056a8f21" title="Click to view the MathML source">{z1,…,zn}⊆Xclass="mathContainer hidden">class="mathCode">, and e
ach scalar
n -tuple
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si10.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=7f1eb2b2fdb15313d6129848871ba4b5" title="Click to view the MathML source">ζ=(ζ1,…,ζn)∈Rnclass="mathContainer hidden">class="mathCode"> that satisfies
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305856&_mathId=si11.gif&_user=111111111&_pii=S0022247X16305856&_rdoc=1&_issn=0022247X&md5=1c195f8e4e472e777093e796833384d2" title="Click to view the MathML source">ζ1+⋯+ζn=0class="mathContainer hidden">class="mathCode">. Notably, these results do not extend to general finite metric sp
aces.