Convex function

Dependencies:

1. Vector Space
2. Convex set

Let $V$ be a vector space over $\mathbb{R}$. Let $S \subseteq V$ be a convex set. Let $f: S \mapsto \mathbb{R}$ be a function.

Then $f$ is defined to be convex iff \[ \forall x \in S, \forall y \in S, \forall \alpha \in [0, 1], f((1-\alpha)x + \alpha y) \le (1-\alpha)f(x) + \alpha f(y) \]

Intuitively, this means that a function $f$ is convex iff the line segment joining $(x, f(x))$ and $(y, f(y))$ lies completely above the function surface.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 6

Transitive dependencies:

1. Group
2. Ring
3. Field
4. Vector Space
5. Convex combination and convex hull
6. Convex set