**Embedding in Orthant**

Slack variables give an embedding of a polytope into the standard *f*-orthant, where *f* is the number of constraints (facets of the polytope). This 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 with *n* vertices (dual to facets), regardless of dimension, as the *image* of the standard -simplex, which has *n* vertices – the map is onto: and expresses points in terms of the *vertices* (points, vectors). The 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.

Read more about this topic: Slack Variable