Properties of distance spaces with power triangle inequalities

Abstract

Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the  triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the  triangle inequality,  relaxed triangle inequality, and   inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of  maximum,  minimum,  mean square,  arithmetic mean, geometric mean, and  harmonic mean as special cases.