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More generally, suppose the optimal value remains constant if the cost coefficient vector $LaTeX: c$ in a linear program is replaced with any of $LaTeX: c + d^1, c+d^2, \ldots , \mbox{ or } c + d^k$ (we could have $LaTeX: k = n$ and let $LaTeX: d^j$ be the j-th coordinate extreme for $LaTeX: c_j$, but that is not necessary). Then, the optimal objective value is the same for $LaTeX: \textstyle c + \sum_{i=1}^k \mu_i d^i$, provided $LaTeX: \mu_i \ge 0$ and $LaTeX: \textstyle \sum_{i=1}^k \mu_i \le 1$
The same applies for convex combination of changes in the right-hand side $LaTeX: b$ (maybe with the origin, which is no change). If the objective value remains optimal at if $LaTeX: b$ is replaced with any of $LaTeX: b + f^1, b + b^2, \ldots, b + f^q$, then it is also optimal for the right-hand side $LaTeX: \textstyle b + \sum_{i+1}^q \beta_i f^i$ so long that $LaTeX: \beta_i \ge 0$ and $LaTeX: \textstyle \sum_{i=1}^q \beta_i \le 1$