这是一份manchester曼切斯特大学ECON20120T/ECON30290T/ECON30320T/ECON60562T 作业代写的成功案例

whence
$$
\sum_{i=1}^{m} \lambda_{i} \cdot x_{i}=\lambda_{i_{0}} x_{i_{0}}+\left(1-\lambda_{i_{0}}\right) \cdot\left(\sum_{j \neq i_{0}} \frac{\lambda_{i}}{\sum_{i \neq i_{0}} \lambda_{i}} \cdot x_{j}\right) .
$$
Since
$$
\sum_{j \neq i_{0}} \frac{\lambda_{i}}{\sum_{i \neq i_{0}} \lambda_{i}} \cdot x_{j}
$$
is a convex linear combination of $m-1$ points of $S$, the induction assumption ensures that it is a point of $S$. Thus $\sum_{i=1}^{m} \lambda_{i} \cdot \boldsymbol{x}_{i}$ can be expressed as a convex linear combination of two points in $S$. From the assumed convexity of $S$, therefore, it belongs to $S$.

ECON20120T/ECON30290T/ECON30320T/ECON60562T COURSE NOTES ：

The assumed concavity of $f(\boldsymbol{x})$ and $g_{i}(\boldsymbol{x}), i=1, \ldots, m$, implies that $A$ and $B$ are convex sets. Corresponding to any two points
$$
\left(\begin{array}{c}
z_{0 i} \
z_{i}
\end{array}\right) \in A \quad(i=1,2),
$$
there exist $x_{i} \in X, i=1,2$, such that
$$
\left(\begin{array}{c}
z_{0 i} \
z_{i}
\end{array}\right) \leqq\left(\begin{array}{l}
f\left(\boldsymbol{x}{i}\right) \ g\left(\boldsymbol{x}{i}\right)
\end{array}\right) \quad(i=1,2)
$$