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
9fd7d5cb26a068936bb1f487c" title="Click to view the MathML source">(X,d) be a given finite ultrametric space with minimum non-zero distance
α. Then the
p -negative type gap
e758d997e59ea" title="Click to view the MathML source">ΓX(p) of
9fd7d5cb26a068936bb1f487c" title="Click to view the MathML source">(X,d) is positive for all
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
808912dc368c5cfb5c361ec1205a75" title="Click to view the MathML source">ΓX(p)/αp is a constant independent of
p . The determination of
ΓX(∞) also leads to new, asymptotically sharp, families of enhanced
p -negative type inequalities for
9fd7d5cb26a068936bb1f487c" title="Click to view the MathML source">(X,d). Indeed, suppose that
bc05e92e30d66e0be2cbe3d617fc" title="Click to view the MathML source">G∈(0,ΓX(∞)). Then, for all sufficiently large
p, the inequality
holds for each finite subset
{z1,…,zn}⊆X, and each scalar
n -tuple
ζ=(ζ1,…,ζn)∈Rn that satisfies
e777093e796833384d2" title="Click to view the MathML source">ζ1+⋯+ζn=0. Notably, these results do not extend to general finite metric spaces.