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?



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