Show that each \(n\) simplex in the barycentric subdivision of \(\Delta^n\) is defined by \(n\) inequalities \(t_{i_0} \leq t_{i_1} \leq \dots \leq t_{i_n}\) in its barycentric coordinates, where \((i_0, \cdots, i_n)\) is a permutation of \((0, \cdots, n)\).
Solution
1. The barycentric subdivision of \(\Delta^n\) has one vertex for each nonempty face of \(\Delta^n\), i.e. for each nonempty subset \(S \subseteq \{0, \ldots, n\}\).
definition of barycentric subdivision.
2. The vertex corresponding to \(S\) is the barycenter of the face spanned by \(\{e_i : i \in S\}\), whose barycentric coordinates are \(t_i = 1/|S|\) for \(i \in S\) and \(t_i = 0\) for \(i \notin S\).
the barycenter of a face.
3. An \(n\)-simplex of the subdivision is spanned by a chain of faces \(S_0 \subsetneq S_1 \subsetneq \cdots \subsetneq S_n\) with \(|S_k| = k + 1\).
a maximal chain of faces gives an \(n\)-simplex of the subdivision.
4. Such a chain corresponds to a permutation \((i_0, \ldots, i_n)\) of \((0, \ldots, n)\), where \(S_k = \{i_0, \ldots, i_k\}\).
3 (each step adds one new vertex).
5. A point in the simplex spanned by the barycenters of \(S_0, \ldots, S_n\) has barycentric coordinates satisfying \(t_{i_0} \ge t_{i_1} \ge \cdots \ge t_{i_n}\).
the barycenter of \(S_k\) has \(t_{i_0} = \cdots = t_{i_k} = 1/(k+1)\) and \(t_{i_{k+1}} = \cdots = t_{i_n} = 0\), so convex combinations of these barycenters have \(t_{i_0} \ge t_{i_1} \ge \cdots \ge t_{i_n}\).
6. Hence each \(n\)-simplex of the subdivision is defined by the \(n\) inequalities \(t_{i_0} \le t_{i_1} \le \cdots \le t_{i_n}\) (up to reversing the order), for a permutation \((i_0, \ldots, i_n)\).
4 and 5.
7. Q.E.D.
6.