Asymmetric norms, cones and partial orders

If X is a real vector space and p an asymmetric norm on X  , the set Cp={x∈X:p(−x)=0}Cp={x∈X:p(−x)=0} is a proper cone in X which induces a partial order on X compatible with the linear structure of X  . Using the norm ps(x)=max⁡{p(x),p(−x)}ps(x)=max⁡{p(x),p(−x)}, a second asymmetric norm can be defined by qp(x)=inf⁡{ps(x+y):y∈Cp}qp(x)=inf⁡{ps(x+y):y∈Cp}. In the case where the partial order induced by CpCp is a lattice order, it is possible to define a third asymmetric norm by p+(x)=p(x+)p+(x)=p(x+), where x+x+ is the positive part of x. The paper investigates the relationships between these three asymmetric norms, with special attention to the case where X is finite-dimensional.