Convex Sets

Topics: Convex Optimization

Definition

A set $\mathcal{C}$ is convex if and only if it contains the line segment connecting any two of its points. That is, if $x,y \in \mathcal{C}$ then their convex combination $\alpha x + (1-\alpha) y \in \mathcal{C}$.

Operations Preserving Convexity

• Intersection: $\mathcal{C}_1 \bigcap \mathcal{C}_2$
• Minkowski sum: $\mathcal{C}_1 + \mathcal{C}_2$
• Translated set: $\mathcal{C} +a$
• Scaled set: $t \mathcal{C} = \{ tx | x \in \mathcal{C}\}$
• Cartesian product: $\mathcal{C}_1 \times \mathcal{C}_2 = \{ (x_1, x_2) | x_1 \in \mathcal{C}_1, x_2 \in \mathcal{C}_2 \}$
• Coordinate projection: $\{x_1 | (x_1, x_2) \in \mathcal{C} \}$
• Image under linear transformation
• Inverse image under linear transformation

References

1. S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge: Cambridge University Press, 2004.