3

The problem is the following.

Let $(X,\|\cdot\|)$ be a normed space and let $C \subseteq X$ be a closed convex set with nonempty interior.

Let be $x\in C$. I define to be a "normal cone to $C$ in $x$" the set: $N_C(x):=\{ f \in X^\ast : \langle f,y-x\rangle \leq 0, \forall y \in C \}$

I would like to prove that if $x \in \partial C$, then $N_C(x)$ has at least a nonzero functional.

I have no idea about where to start... I've tried something with the geometric forms of Hahn Banach for separating the convex sets, but probably is not the right way to follow.

Any help is appreciated!

Benzio
  • 2,097

1 Answers1

2

I found the answer myself after I started the bounty and I lost the reputation. Now I'm feeling stupid!

Well, if the interior $int(C)$ is convex, open (by definition of interior), and nonempty, and $x_0$ is a point such that $x_0 \notin int(C)$, then by the geometric form of Hahn Banach it's possible to find a functional $f \in X^\ast$ such that $f(y)<\alpha \leq f(x_0)$ for every $y \in int(C)$. Then $f(y-x)=\langle f,x-y\rangle < 0$ for every $y \in int(C) \neq \emptyset$. Then $f \neq 0$.

The fact that the bounty is not refunded if you find the answer by yourself may discourage some people in trying to find the answer after the bounty is placed. Do you agree?

Benzio
  • 2,097