Consider a pointed closed convex polyhedral cone $C$ in $\mathbb R^n$.
Let $S_1,..,S_n$ be hyperplanes in $\mathbb R^n$.
Consider $S=C \cap \left(\bigcap\limits_{i=1}^n S_i\right)$.
If $S$ is compact and nonempty, is it true that the Euler charcteristic is equal to 1?
In the simple case of $C$ being the positive orthant with one hyperplane given by $\sum_{i=1}^n x_i = 1$, I know that the answer is affirmative since $S$ will simply be a simplex.
If the claim is true in general or for some special case, please provide me a reference that states this.
Thank you.
Edit: To have some meaningful sub-example,
Consider $C$ to be the closed positive orthant
Let $S_1$ be given by $\sum_{i=1}^n \alpha_i x_i = M_1$
$S_2$ be given by $\sum_{i=1}^n \beta_i x_i = M_2$, where $\alpha_i,\beta_i\ge 0, M_1,M_2 >0$.
Is it true in this case?
The intersection of finitely many affine hyperplanes in $\mathbb R^n$ is a proper affine subspace $U\subseteq \mathbb R^n$. So you can describe $S$ equivalently as $C\cap U$, where $C$ is a pointed closed convex polyhedral cone and $U$ a proper affine subspace. The intersection of a polyhedral cone with an affine subspace is either empty or a polyhedron. In the non-empty case you have $\chi(S)=1$ since polyhedra are contractible.