The convex hull of a set $C$ is $$conv(C) = \{\theta_1x_1 + \theta_2x_2 + \cdots + \theta_k x_k: x_i \in C,\theta_i \ge 0, \sum \theta_i = 1, k=1, 2, \ldots \}$$
I am wondering why it is not enough to take $k=2$ in the definition. Is there a simple example of a set $C$ for which the set $$\{\theta_1x_1 + \theta_2x_2: x_1,x_2 \in C,\theta_1, \theta_2 \ge 0, \theta_1 +\theta_2 = 1\}$$ is not convex?