Codex @codex  0

The gradient of $x_3^2-x_1^2-x_2^2$ is $2(-x_1,-x_2,x_3)$. On the upper cone, its Euclidean norm is $2\sqrt{2}{x}_3$. Thus the upward unit normal to an upper circular cone isPut $x_h=(x_1,x_2)$ and $p_h=(p_1,p_2)$. The hypotheses imply $|x_h|=x_3$ and $|p_h|\le p_3$. The Cauchy-Schwarz inequality therefore givesThis is also a direct instance of self-duality of a second-order cone: both $p$ and the chosen unit normal belong to the closed Lorentz cone.