Dual Cones
Every cone casts a shadow: the directions that lean the same way as all of it.
The idea
The dual cone of a cone $K$ — a set closed under scaling by any nonnegative number — collects the directions $y$ that make an angle of at most $90^{\circ}$ with every point of $K$. A direction $y$ makes such an angle with a point $x$ exactly when $x^{T}y \ge 0$, so
$K^{*} = \{y : x^{T}y \ge 0 \text{ for all } x \in K\}.$
The condition has a reading in terms of halfspaces. Fix $y$. The set $\{x : y^{T}x \ge 0\}$ is the halfspace on one side of the hyperplane through the origin perpendicular to $y$, and $y \in K^{}$ says precisely that this halfspace contains the whole of $K$. Tilting $y$ far enough pushes part of $K$ through the hyperplane, and $y$ leaves $K^{}$.
The halfspace reading is also what makes the construction useful. The membership condition "$x$ lies in $K
quot; is awkward to compute with, and the dual cone converts it into linear inequalities $x^{T}y \ge 0$, one for each $y \in K^{*}$ — a single geometric condition traded for a family of linear ones.Ways to work on it
- Walkthrough. Dualize a wedge in the plane, see how the angles respond, and meet self-dual cones.
- Proof. Why a dual cone is always convex and closed, even when the original cone is neither.
- Practice. Duals of wedges, of lines, and of single rays.
- Hardest. Compute the duals of norm cones.
Not sure where to start? Take the ten-question placement test.