Suppose that $C$ is a convex set such that $0\in C\subset\mathbb{R}^n$ and $$\forall x\in \mathbb{R}^n\backslash\{0\},\exists\{x_k\}_k\subset C\text{ s.t. }\lim_{k\to\infty} \frac{x_k}{\|x_k\|}=\frac{x}{\|x\|}$$ The question is whether it can be concluded that there exists a small enough $\epsilon$ satisfying $$ B_\epsilon \subset C$$ So far I didn't come up with any good idea. The convexity seems to play an important role here since without it the statement is wrong.
Asked
Active
Viewed 21 times
1
-
This problem is solved in the proof link which uses seperation theorem of hypersurfaces though seems not so elementary from definition. – Oolong Milktea Apr 03 '23 at 14:03
-
Did you see this? https://math.stackexchange.com/questions/4305234/every-nonzero-direction-at-x-is-feasible-iff-x-lies-in-the-interior-of-th?rq=1 – MathFont Apr 03 '23 at 18:23
-
@MathWonk. Feasible directions in the question you posted are different in definition from the Clarke's tangent cone in my question, and this is stronger than that. – Oolong Milktea Apr 04 '23 at 05:12