Convex functions

Ideally we want our cost/loss function to be convex(concave).

First we give the definition of a convex set: A set $C$ in $\mathbb{R}^n$ is said to be convex if, for all $x$ and $y$ in $C$ and all $t \in (0,1)$ , the point $(1 − t)x + ty$ also belongs to C. Geometrically this means that every point on the line segment connecting $x$ and $y$ is in $C$ as discussed below.

The convex subsets of $\mathbb{R}$ are the intervals of $\mathbb{R}$. Examples of convex sets of $\mathbb{R}^2$ are the regular polygons (triangles, rectangles, pentagons, etc...).

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