Editing Barycentric coordinate system (section)

=== Abstraction ===
More abstractly, generalized barycentric coordinates express a convex polytope with ''n'' vertices, regardless of dimension, as the ''image'' of the standard <math>(n-1)</math>-simplex, which has ''n'' vertices – the map is onto: <math>\Delta^{n-1} \twoheadrightarrow P.</math> 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 except when P is a simplex.

[[Dual linear program|Dual]] to generalized barycentric coordinates are [[slack variable]]s, which measure by how much margin a point satisfies the linear constraints, and gives an [[embedding]] <math>P \hookrightarrow (\mathbf{R}_{\geq 0})^f</math> into the ''f''-[[orthant]], where ''f'' is the number of faces (dual to the vertices). This map is one-to-one (slack variables are uniquely determined) but not onto (not all combinations can be realized).

This use of the standard <math>(n-1)</math>-simplex and ''f''-orthant as standard objects that map to a polytope or that a polytope maps into should be contrasted with the use of the standard vector space <math>K^n</math> as the standard object for vector spaces, and the standard [[affine hyperplane]] <math>\{(x_0,\ldots,x_n) \mid \sum x_i = 1\} \subset K^{n+1}</math> as the standard object for affine spaces, where in each case choosing a [[linear basis]] or [[affine basis]] provides an ''isomorphism,'' allowing all vector spaces and affine spaces to be thought of in terms of these standard spaces, rather than an onto or one-to-one map (not every polytope is a simplex). Further, the ''n''-orthant is the standard object that maps ''to'' cones.