Convex relaxation for the complement of Lorentz cone

Question

Is it possible to obtain a convex relaxation for $$ \{ (x,t): t \le \|x\|_2\} \in \mathbb{R}^{d+1} $$ where $x \in \mathbb{R}^d$ and $\|x\|_2$ is the usual Euclidean norm, by moving to higher dimensions and projecting back (the so-called lifting)?

Answer 1

The cone $\{ (x,t): t \le \|x\|_2\} $ cannot be the image of a convex set under a linear map. But it can be realized as the image of the halfspace $\{ (x,t): t \le 0 \} $ set under the following nonlinear map: $F(t,x) = (t+\|x\|_2,x)$. The map is a bijection, the inverse being $F^{-1}(t,x) = (t-\|x\|_2,x)$. It is a kind of shear deformation.