Let $\Delta_{n-1}$ denote the standard probability simplex $$\{(x_1,\cdots, x_n):\sum_{i}^n x_i=1, x_i\geq1 \text{ for all } i=1,\cdots,n\}.$$ Any $d$-dimensional polytope with $n$ facets is affinely isomorphic to an intersection of the $\Delta_{n-1}$ with an affine subspace of dimension $d$.
Is it also true that every $d$-polytope with $n$ facets is combinatorially isomorphic to the intersection of $\Delta_{n-1}$ with a linear space of dimension $d$? In other words, is every such polytope combinatorially isomorphic to one of the form $\{x\in\mathbb{R}^n: Bx=0, \sum_{i=1}^n x_i=1, x\geq 0\}$ for some $B\in\mathbb{R}^{(n-d)\times n}$?
No.
The $(n-1)$-dimensional simplex cannot be written as the intersection of $\Delta_{n-1}$ with an $(n-1)$-dimensional linear subspace.