Boyd's (Chapter 2) exercise 2.34 (c):
$K_{lex} = \{ 0\} \cup \{ x \in \mathbb{R}^n | x_1= \dots = x_k, x_{k+1} > 0\}$, for some $k, 0 \leq k < n$
and the dual cone $K^*_{lex}$ is defined as:
$K^*_{lex} \neq \mathbb{R}_{+e_1} = \{(t,o, \dots, 0)| t\geq 0 \}$
my question is, what does it mean by $\mathbb{R}_{+}e_1$?
And also why $K^*_{lex} \neq \mathbb{R}_+e_1 = \{(t_1,t_2, \dots, 0)| t_1, t_2 \geq 0 \}$, since that satisfies the condition $y^Tx \geq 0$ for all $x \in K^*_{lex}$ too?
$\mathbb{R}_{+}e_1 = \{ (t,0,...,0)^T \mid t\geq 0\}$ is the set of vectors where the first element is non-negative and the rest are zeros.
$\{ (t_1,t_2,0,...,0)^T \mid t_1,t_2 \geq 0\}$ is not in $K^*_{lex}$. Counter example: Let $x = (t_1,t_2,0,...,0)^T$ and $y = (t_2,-t_1-1,0,...,0)^T \in K_{lex}$. Then, $y^Tx=-t_2 < 0$.
I think you are not fully understanding what the lexicographic cone is. "The lexicographic cone $K_{lex}$ is just the set of points such that there is at least one non-zero coordinate, and the first such coordinate is positive," said by Jorge Fernández in this post.
Hope this helps. :)