operations that preserve fuction convexity

non-negative scaling sum: f_{1}+ f_{2} is convex if f_1, f_2 is convex infinite sums: \sum_{i=1}^{\infty} f_{i} is convex integral: if f\left(x,a\right) is convex in x, \int_{a \in A} f\left(x,a\right) \dd{a} is convex pre-composition with affine function: f\left(Ax + b\right) is convex if f is convex pointwise maximum: f_{1}, …, f_{m} is convex, then f\left(x\right) = \max \left(f_{1} \left(x\right)\dots f_n \left(x\right)\right) is convex supremum: if f\left(x,y\right) is convex in x far each \text{sup}_{y \in Y} f\left(x,y\right) partial minimization: f\left(x\right) = \text{inf}_{y \in C} f\left(x,y\right) (find the smallest value of f over y \in C, or the point at which its approached) perspective of convex function is convex \text{persp}\left(a,b\right) = b f\left(\frac{a}{b}\right) conjugate function of any function is convex composition with scalar functions g : \mathbb{R}^{n} \to \mathbb{R}, h: \mathbb{R} \to \mathbb{R}, and let f = h\left(g\left(x\right)\right) = h \odot g composition f is convex if: g convex, h convex, extended-value extension \tilde{h} non decreasing g concave, h convex, extended-value extension \tilde{h} non increasing composition f is concave if: g concave, h concave, extended-value extension \tilde{h} non decreasing g convex, h concave, extended-value extension \tilde{h} non increasing general composition rule that preserve convexity composition of g : \mathbb{R}^{n} \to \mathbb{R}^{k}, and h: \mathbb{R}^{k} \to \mathbb{R} is f\left(x\right) = h\left(g\left(x\right)\right) = h\left(g_{1}\left(x\right), \dots, g_{k}\left(x\right)\right) f is convex if h is convex and for each i, one of the the following holds: g_{i} convex, \tilde h nondecreasing in its i th element g_{i} concave, \tilde h nonincreasing in its i th element g_{i} affine examples sum of the r largest elements of a set is convex since we can multiply them with many-hot selectors which gives you combinations an and then max them together