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I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone must have an non-negative inner product with any point in its corresponding cone.

How does that bring a diamond shaped $L-1$ norm to a square shaped $L-\infty$ norm?

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Let $C$ be a cone and $C^*=\{y:\langle x,y\rangle\ge 0 \ \forall x\in C\} $ its dual cone. If a point $y$ satisfies $\langle x,y\rangle\ge 0$ for all extreme rays of $C$, then it satisfies this inequality for all rays of $C$. Therefore, we can restrict attention to the extreme rays of $C$. Each of these rays determines a half-plane $\{y:\langle x,y\rangle\ge 0\}$.

The above may be easier to visualize for bounded convex sets than for cones. (The geometry of a cone is determined by its base.) For example, the $\ell_1$ unit ball in $n$ dimensions is diamond-shaped with $2n$ extreme points $\pm e_j$. The half-planes corresponding to these extreme points form $2n$ faces of the $\ell_\infty$ ball, which is a cube.

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