Algebra and Geometry | Part IA, 2006

Consider the cone KK in R3\mathbb{R}^{3} defined by

x32=x12+x22,x3>0.x_{3}^{2}=x_{1}^{2}+x_{2}^{2}, \quad x_{3}>0 .

Find a unit normal n=(n1,n2,n3)\mathbf{n}=\left(n_{1}, n_{2}, n_{3}\right) to KK at the point x=(x1,x2,x3)\mathbf{x}=\left(x_{1}, x_{2}, x_{3}\right) such that n30n_{3} \geqslant 0.

Show that if p=(p1,p2,p3)\mathbf{p}=\left(p_{1}, p_{2}, p_{3}\right) satisfies

p32p12+p22p_{3}^{2} \geqslant p_{1}^{2}+p_{2}^{2}

and p30p_{3} \geqslant 0 then

pn0\mathbf{p} \cdot \mathbf{n} \geqslant 0

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