Second-order cone

Definition

The set in mathbf{R}^{n+1}

 mathbf{K}_{n} :=left{ (x,y) in mathbf{R}^{n+1} ~:~ y ge |x|_2 right}

is a convex cone, called the second-order cone.

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Example: The second-order cone is sometimes called ‘‘ice-cream cone’’. In mathbf{R}^3, it is the set of triples (x_1,x_2,y) with

 y^2 ge x_1^2+x_2^2, ;; y ge 0.

The blue circle corresponds to the set

 left{ (x_1,x_2,y) ~:~ y = 1 ge sqrt{x_1^2+x_2^2} right}.

Proof of convexity

The fact that mathbf{K}_{n} is convex can be proven directly from the basic definition of a convex set. Alternatively, we may express mathbf{K}_{n} as an intersection of half-spaces, as follows.

From the Cauchy-Schwartz inequality, we have

 y ge |x|_2 Longleftrightarrow forall : u, ;; |u|_2 le 1 ~:~ y ge u^Tx ,

we have

 mathbf{K}_{n} = bigcap_{u ::: |u|_2 le 1} left{ (x,y) in mathbf{R}^{n+1} ~:~  y ge u^Tx right} .

Each one of the sets involved in the intersection is a half-space.