UTM Ideals Varieties and Algorithm - Chapter 1 Section 3 Exercise 14

Problem:

Solution:

Part a is obvious. The given formula is the parametric line and by definition it is in the convex set $C$.

Part a is the basis of the hypothesis. Consider it is true for $n = k$, we have

$\begin{eqnarray*} & & \sum\limits_{i=1}^{k+1} t_i \left(\begin{array}{c} x_i \\ y_i \end{array}\right) \\ &=& \sum\limits_{i=1}^{k} t_i \left(\begin{array}{c} x_i \\ y_i \end{array}\right) + t_{k+1}\left(\begin{array}{c} x_{k+1} \\ y_{k+1} \end{array}\right) \\ &=& (\sum\limits_{i=1}^{k} t_i)\sum\limits_{i=1}^{k} \frac{t_i}{\sum\limits_{i=1}^{k} t_i} \left(\begin{array}{c} x_i \\ y_i \end{array}\right) + t_{k+1}\left(\begin{array}{c} x_{k+1} \\ y_{k+1} \end{array}\right) \end{eqnarray*}$

The inner sum is actually a point in $C$ because it is a convex combination.The outer sum is also a point in $C$ because that is also a convex combination!