Let $S$ be a set of points in $\mathbf{R}^n$:
$$S=\{ (x_1,\dots,x_m) \ | \ x_i \in \mathbf{R}^n \ ; i=1,\dots, m \}$$
How do we define and find the dual space or dual polytope of $S$?
Let $S$ be a set of points in $\mathbf{R}^n$:
$$S=\{ (x_1,\dots,x_m) \ | \ x_i \in \mathbf{R}^n \ ; i=1,\dots, m \}$$
How do we define and find the dual space or dual polytope of $S$?
Take an arbitrary point for center and any sphere around that, and then invert your points wrt. to that sphere along the ray from that center to those. Next errect at those inverted points the hyperplanes. These then are the mutual dual facets to those points.
Esp. when you had for the polytope the convex hull of that set of points, then you'd get for the dual polytope the intersection kernel of those halfspaces, defined by the hyperplanes and containing the center point.
--- rk
Edit:
The main issue here is to derive a hyperplane as the dual of a point. This is being done by means of spherical reciprocation. Let's consider the point $P=(p,0,...,0)$. Next consider e.g. the unit sphere around the origin. Then the reciprocal point $P'$ would be located at $(\frac{1}{p},0,...,0)$ and accordingly the hyperplane through that point and orthogonal to that very ray would by given by $x_1=\frac{1}{p}$.
--- rk