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It is well known that an ordered normed vector space $X$ with normal cone$X_+$ has an order-preserving norm that is equivalent to the original norm.Such an equivalent order-preserving norm is given by \begin{equation} \sharp x\sharp = \max \{ d(x, X_+), d(x, - X_+)\}, \qquad x \in X. \end{equation} Thispaper explores the properties of this norm and of the half-norm $\psi(x) =d(x,-X_+)$ independently of whether or not the cone is normal. We use $\psi$ toderive comparison principles for the solutions of abstract integral equations,derive conditions for point-dissipativity of nonlinear positive maps, compareCollatz-Wielandt numbers, bounds, and order spectral radii for boundedhomogeneous maps and give conditions for a local upper Collatz-Wielandt radiusto have a lower positive eigenvector.

Comparison of spectral radii and Collatz-Wielandt numbers for  homogeneous maps, and other applications of the monotone companion norm on  ordered normed vector spaces