Slack variable
A non-negativity constraint on the slack variable is also added.Slack variables give an embedding of a polytopeThis map is one-to-one (slack variables are uniquely determined) but not onto (not all combinations can be realized), and is expressed in terms of the constraints (linear functionals, covectors).Slack variables are dual to generalized barycentric coordinates, and, dually to generalized barycentric coordinates (which are not unique but can all be realized), are uniquely determined, but cannot all be realized.Dually, generalized barycentric coordinates express a polytope withvertices (dual to facets), regardless of dimension, as the image of the standardThe map is one-to-one if and only if the polytope is a simplex, in which case the map is an isomorphism; this corresponds to a point not having unique generalized barycentric coordinates.