A contribution to a geometric understanding of p-norms

There is growing interest for L^p minimization. Nevertheless, the relationship between solutions for various values of p is rarely considered. This paper is a contribution to a better understanding of the geometry of these solutions in the simple case of a system with three equations and two unknowns. Using barycentric coordinates in a certain triangle, the locus of solutions is shown to be a smooth curve, with a very simple parameterization linked to the duality in spaces L^p, highlighting certain particular “centers”, such as the mean squares solution which is the Lemoine (symmedian) point of the triangle.