Boyd's Exercise: 2.34 Lexicographic cone is defined as:
$K_{lex} = \{ 0\} \cup \{x \in \mathbb {R}^n | x_1 = \dots = x_k = 0, x_{k+1} > 0, \text{ for some } k, 0 \leq k < n \}$
The solution is given as:
Take $(\epsilon, -1, 0, \dots, 0) \in K_{lex}$ for all $\epsilon > 0$, but not for $\epsilon = 0$.
But it does not make sense to me. Could anyone please explain a little bit more or perhaps any geometrical explanation of how the lexicographic cone would look like? Thanks.
The lexicographic cone $K$ is just the set of points such that there is at least on non-zero coordinate, and the first such coordinate is positive.
To see it is not closed you just need to notice that the point $(0,-1,0,\dots, 0)$ is a limit point of $K$ but it is not in $K$. To see why it is a limit point notice that $(\epsilon,-1,0,\dots,0)$ is always an element of $K$ for $\epsilon>0$ because its first non-zero coordinate is positive.
As for how to visualize the geometric cone you can see it as a union of $n$ half-hyperplanes of decreasing dimension.
The first one is the hyperplane of points $(a_1,a_2,\dots,a_n)$ such that $a_1\in \mathbb R^+$. The second one is the hyperplane of points $(0,a_2\dots, a_n\}$ such that $a_2>0$ etc.