I've found here http://math.stanford.edu/~conrad/diffgeomPage/handouts.html a very interesting paper on Stokes theorem for manifolds with corners. So I've decided to read a paper on manifolds with corners which is also available there: http://math.stanford.edu/~conrad/diffgeomPage/handouts/corners.pdf
But the first papagraph turned out to be very problematic because of the notation. I've tried browsing previous articles but nothing like that came up.
Here is what bothers me:
Let $W$ be an $m$-dimensional $\mathbb{R}$-vector space, $m \ge 1$. For $ 1 \le k \le m$ a $k$-sector in $W$ is a non-empty subset of the form $$ \Sigma = \{w \in W | `l_1(w) \ge c_1, . . . , l_k(w) \ge c_k\}$$ with $c_1, . . . , c_k \in \mathbb{ R}$ and linearly independent $l_1, . . . , `l_k \in W^{\vee}$.
A $0$-sector is $ \Sigma = W$. A sector $ \Sigma \subset W$ is a $k$-sector for some $k$. If $w \in W$ is a point, then the translation $w + \Sigma $ also a $k$-sector: we use the same $i$’s but replace $c_i$ with $c_i + `l_i(w)$.
Could you tell me what $W^{\vee}$ can mean here?
I would also be very grateful to you, if you could explain to me what a $k$-sector is.
It means the dual space of $W$. Another notation is $W^\ast$.