Let $x_1, x_2 \in \Bbb R^d$ be two rays that define a ‘noisy’ convex cone $$ C := \{ \lambda x_1 + (1 - \lambda) x_2 + \epsilon \mid \lambda \ge 0 \}, $$ for some vector $\epsilon \in \Bbb R^d$ with a small norm. Is it enough to assume that $x_1 \cdot x_2 < 0$ in order to say that $\forall p \in C, -p \notin C$?
I would appreciate it if you’ll have any references to support your answer.
Tnx.
Let $d=1$, let $x_1=1$ and $x_2=-1$, $$\lambda x_1+(1-\lambda)x_2+\epsilon =\lambda -(1-\lambda)+\epsilon = 2\lambda -1+\epsilon \ge -1 +\epsilon $$
Hence if $0<\epsilon<1$, we have both $-1+\frac{1+\epsilon}2=\frac{\epsilon-1}2$ and $\frac{1-\epsilon}2 \in C$. To be even clearer, we can take $\epsilon = 0.5$, that is we have $-0.25$ and $0.25$ are both in $C$.
For $d>2$, to construct similar counter example, let $\epsilon = 0.25e_1$ where $e_1$ is the first unit basis, $x_1=e_1$ and $x_2=-e_1$. Then $\lambda e_1-(1-\lambda)e_1+0.25e_1=(2\lambda -0.75)e_1$ and we can repeat the same argument.
Remark: is the definition found in a book? Currently, it corresponds to a half line. I think it's more natural to define it by $C=\{\lambda_1 x_1 + \lambda_2x_2 + \epsilon |\lambda_1, \lambda_2 \ge 0\}$