Does anyone know a counterexample to show that a weighted sum of convex sets is not necessarily convex, unless our coefficients are positive?
A weighted sum for me is defined as: $$\alpha C_1 + \beta C_2 = \{ y : y = \alpha x_1 + \beta x_2, x_1 \in C_1, x_2 \in C_2, \alpha , \beta \in \mathbb{R} \}$$.
It is easy to prove this is convex in the case of positive constants, however, the proof does not necessarily take account of the fact that the constants are positive. What is a counterexample of the case for any real coefficients $\alpha$, $\beta$?
Weighted sums of convex sets are always convex, even if the factors are negative.
Note that $\alpha C$ is convex for every $\alpha \in \mathbb{R}$, if $C$ is convex. Then use the fact that the Minkowski sum of convex sets is convex.