Normal cone of sphere

Question

I want to to compute the normal cone to sphere $C=\{x\in\mathbb R^{n}:||x||=1\}$. The definition of normal cone to a set $C$, is the set $$N_C(x)=\{g\in \mathbb R^n: g^{T}(u-x)\le 0, \mbox{ for all } u\in C\},$$ where $x\in C$. I see that $$g^{T}u\le ||g||$$ using Cauchy-Schwartz inequality, but I can't find another expression to characterize the values of $g$. How can I find a more xplicit form of $g$?

Answer 1

This particular definition of the normal cone is only for convex sets. Since the sphere is not convex, using this definition doesn't really make sense. (e.g. in $\mathbb{R}^2$, take $x_1=(0,1)$ and $x_2=(0,-1)$. Both $x_1$ and $x_2$ reside in $C$, however their convex combination $(x_1+x_2)/2=(0,0)$ is not in $C$, since its norm is not $1$.)

For a general definition of a normal cone, check out the book by Rockafellar/Wets.

By the way, if we take the convex hull of this set, you get the ball consisting of all $x\in\mathbb{R}^n$ such that $\|x\|\leq 1$. Computing the normal cone to the ball of any center and radius is a classical exercise in convex analysis :)