Optimal lower bounds on the maximal p-negative type of finite metric spaces

TL;DR

This paper establishes elementary lower bounds on the maximal p-negative type of finite metric spaces, depending on their size and diameter, and demonstrates their optimality in various cases.

Contribution

It provides the first elementary derivation of optimal lower bounds on the supremal p-negative type for finite metric and semi-metric spaces.

Findings

Lower bounds depend only on space size and diameter

Bounds are often best possible under certain conditions

Theory extends to finite semi-metric spaces without changes

Abstract

This article derives lower bounds on the supremal (strict) p-negative type of finite metric spaces using purely elementary techniques. The bounds depend only on the cardinality and the (scaled) diameter of the underlying finite metric space. Examples show that these lower bounds can easily be best possible under clearly delineated circumstances. We further point out that the entire theory holds (more generally) for finite semi-metric spaces without modification and wherein the lower bounds are always optimal.