In the study of convexity and convex polyhedra there are three (related) important notions of duality
1) Polar duality
This is a map assigning to every convex set $K$ containing the origin its polar dual: $K^*$ which is the set of all points whose inner product with every point in $K$ is at most 1.
On polytopes it induces an order reversing map on the face lattices. This operation has subtle relations to mirror-symmetry and Koszul duality.
2) Gale transform
This is an operation to move from n points in $R^d$ to n points in $R^k$ where k=n-d-1. It is especially useful if the original n points are in convex position to study the convex polytope they define. (Web-sources: 1, 2; 3; 4)
3) Linear programming duality
This is an operation to move from a linear programming problem to a dual problem which have the same solution.
