Convex set

A set $$C$$ is convex. if the line segment between two points in $$C$$ lies in $$C$$, i.e., if $$\theta x_1 + (1-\theta) x_2 \in C$$ and $$0 \le \theta \le 1$$.

Convex Combination
A linear combination of points $$\sum_{i} \theta_i x_i$$ where $$\sum_i \theta_i = 1$$ and $$\theta_i \ge 0$$ is a convex combination

Operations that preserve set convexity
and the inverse image of $$S$$ under $$f$$, $$f^{-1}(S) = \{x \mid f(x) \in S \}$$ are both convex if $$S$$ is convex.
 * '''Intersection}: Convexity is preserved under intersection; the intersection of even infinite convex sets is convex.
 * Affine functions. Let $$f$$ be an affine function, \ie $$f(x)=Ax+b$$. Then the image of $$S$$ under $$f$$, $$f(S) = \{f(x) \mid x \in S\}$$


 * Cartesian product. Define $$S:=S_1\times S_2$$ for two convex sets $$S_1, S_2$$. Then, $$S$$ is convex.
 * Sum The sum $$S$$ of two convex sets $$S_1, S_2$$, $$S=\{x_1 + x_2 : x_1 \in S_1, S_2 \in x_2\}$$ is convex.


 * Linear-fractional and perspective functions.

* Perspective function is the $$R^{n+1} \to \mathbf R^n $$ function $$P(x,t) = x/t$$ with domain $$\textbf{dom}{P} = \mathbf R ^{n+1} \times \mathbf R_{++}$$. That is, the perspective function normalizes the input vector so the last element is one, and then drops this last element.

If a set $$C \subseteq \setop{dom}{P}$$ is convex, then its image under $$P$$ is also convex. * Linear-fractional function is the composition $$P \circ g$$ of a perspective function $$P$$ with an affine function $$g$$.

It is easy to show that linear-fraction functions preserve convexity: If $$S$$ is convex, then its image $$g(S)$$ under $$g$$ will be convex, then its image under perspective will also be convex.