A Norm Reducing Property for Isotonized Cauchy Mean Value Functions

Abstract

We consider functions $\alpha(\bullet)$ and $\hat{\alpha}(\bullet)$ on a finite set $S$ which correspond to a function $M(\bullet)$ on the nonempty subsets of $S$ which has the Cauchy mean value property (i.e., $M(A + B)$ is between $M(A)$ and $M(B)$ whenever $A$ and $B$ are nonempty disjoint subsets of $S$). $\hat{\alpha}(\bullet)$ is isotone with respect to a partial ordering on $S$ and is equal to $\alpha(\bullet)$ when $\alpha(\bullet)$ is isotone. It is shown that $\hat{\alpha}(\bullet)$ has the following norm reducing property: $\max_{s\in S} |\hat{\alpha}(s) - \theta(s)| \leqq \max_{s\in S} |\alpha(s) - \theta(s)|$ for all isotone $\theta(\bullet)$. Computation algorithms for $\hat{\alpha}(\bullet)$ are discussed and the norm reducing property is shown to give consistency results in several isotonic regression problems.