# Slack Variables and Duality in Convex Optimization

In context of convex optimization the slack variable $$\vec{s} \ge 0$$ can be used to convert inequality $$A \vec{x} \le \vec{b}$$ to the equality $$A \vec{x} + \vec{s}= \vec{b}$$.

Now in wikipedia is claimed that if we regard slack variables abstractly as a datum for embedding of a polytope $$P \hookrightarrow (\mathbf{R}_{\geq 0})^f$$ (where $$f=$$ number of constraints; explicitly I guess the map should be given as $$x \mapsto s(x)$$ where $$x \in P=P(A,b)=$$ polytope determined by inequalities $$Ax \le b$$ and $$s(x)$$ satistied $$Ax+s(x)=b$$),
then slack variables may be recognized to be dual to generalized barycentric coordinates via realization of a polytope as image of onto map $$\Delta^{n-1} \twoheadrightarrow P$$.

Question: In which sense is this a duality in sense of duality principle in convex optimization? Or at least how is it related to it?

Going a step back, firstly what is the "nature" of this putative duality? Is it related to duality from convex optimization (as linked above), or is it just a "naive/vague" duality in the sense that just that one is embedded in a standard simplex, the other is projected from such and that's regarded as "dual", and that's it, nothing deeper?