in completing my thesis I have reached a momentary impass. I am trying to solve an exercise given in the book "Foliations II" by Candel and Conlon. In particular, Exercise 10.4.1, and I can't seem to get through it.
Here is what I have to solve and can't seem to manage: Let $V=\{\lambda_1,\dots,\lambda_p\}\subset\partial B_\xi$ be an affinely independent set, where $\xi$ is the Thurston norm and $\partial B_\xi$ is the boundary of the unit ball of this norm $B_\xi=\{w\in H_2(M,\partial M;\mathbb{R})|\xi(w)=1\}$. Prove that the affine $p$-simplex $\Delta_p$, spanned by $V$, is a subset of $B_\xi$. Generally, $\Delta_p$ is not a subset of $\partial B_\xi$, but if an interior point $\lambda$ of $\Delta_p$ has norm $\xi(\lambda)=1$ prove that $\Delta_p\subset\partial B_\xi$.
I will be using this for the case where $M$ is the complement in $3$-space of a link or knot, but I think this should work in general for $3$-manifolds. Any advice, or elegant solutions?? Thanks in advance, Paul
Using subadditivity we get for $x$ in $\Delta$ that $\xi x = \xi(\sum \lambda_i v_i) \le \sum \lambda_i \xi(v_i) =1$.
For the second part note that if we restrict $\xi$ to the interior $int \Delta \to \mathbb R$, the set $\xi^{-1}(1)$ is closed and non empty. It is also obviously open (use that $x$ in the interior can only be written st all $\lambda_i$ are non zero) and hence the above restriction factors through the trivial space $\{1\}$ and hence through the boundary.