Polarity and dual convex bodies

For a set K in real d-space, its polar consists of all vectors y whose inner product with every point of K is at most one. In this lesson K is a compact convex body with zero in its interior. The polar is then also compact, convex, full dimensional, and contains zero in its interior. Convexity follows because the definition is an intersection of halfspaces, one for each point of K.

The interior assumption does real work. A small ball around zero inside K bounds the norm of every polar vector, while boundedness of K supplies a small ball inside the polar. If zero lies only on the boundary, the polar can be unbounded: the interval [0, 1] has polar (−∞, 1] in one dimension. If the chosen origin changes, first translate K to that origin and recompute; ordinary polarity is not translation invariant.

If K is contained in L, a vector satisfying every inequality from L also satisfies those from K. Therefore the polar of L is contained in the polar of K. Scaling K outward by a positive factor scales its polar inward by the reciprocal factor. More generally an invertible linear map A sends the polar to A inverse transpose applied to the old polar, because the pairing with Ax equals the pairing of x with A transpose y.

Applying polarity twice returns K under our closed-convex, origin-containing hypotheses. One inclusion follows directly from the definition; the reverse follows by separating a point outside K from K. Without convexity and closure, double polarity generally produces a closed convex hull containing zero rather than the original set. This distinction parallels optimization duality: inequalities can capture a convex envelope even when the initial collection of points was not convex.

For a full-dimensional polytope containing zero in its interior, a proper nonempty face F corresponds to the dual face where the inner product with every point of F equals one. Inclusion of faces is reversed, and dimensions add to d − 1. Consequently vertices correspond to facets, while edges in three dimensions correspond to edges. The f-vector therefore reverses, but this statement concerns the polar’s combinatorics rather than arbitrary geometric dual drawings.

Support functions make calculations concrete. The support of a centered axis-aligned rectangle is the sum of its half-widths times the absolute values of the dual coordinates. Setting that expression at most one gives a diamond. A long primal direction becomes a short dual direction. This reciprocal relation is useful for norm balls and optimization constraints, but it depends on the chosen pairing and normalization; a polar is not obtained by merely swapping x and y labels.