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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.

Web sources (1; 2; 3; 4)

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.

(We sources: 1; 2; 3;)

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.