Is the set of binding constrains of linear programming unique?

Question

Consider the following linear program $\min c^Tx$ s.t. $Ax\leq b$, where $A\in\mathbb{R}^{m\times n}$ We know that the optimal solutions $x^*$ of the above linear program are not unique. And we also know that for an optimal solution $x^*$, some constraints are binding, defined as $B=\{i\in[m]|A_ix=b_i\}$, where $A_i$ is the $i$th row of matrix A. My question is that is the set of binding constraints $B$ unique for a given linear program? how to prove that or disapprove that?

Answer 1

Different basic feasible solutions will have different sets of binding constraints. That is, a basic feasible solution is determined by the set of binding constraints for it.