Convex function

Let $𝑋$ be a convex subset of $ℝ 𝑛$ . A function $𝑓 : 𝑋 \to ℝ$ is said to be convex iff its epigraph (the set of points above its Graph set) is a convex subset of $ℝ 𝑛 + 1$ . #m/def/anal Equivalently, for all $𝑡 \in [0, 1]$ and $𝑥 1, 𝑥 2 \in 𝑋$ ,

𝑓 (𝑡 𝑥 1 + (1 - 𝑡) 𝑥 2) \leq 𝑡 𝑓 (𝑥 1) + (1 - 𝑡) 𝑓 (𝑥 2)

i.e. the secant lies above the graph. This is sometimes referred to as Jensen's inequality for two points. Such a function is strictly convex iff

𝑓 (𝑡 𝑥 1 + (1 - 𝑡) 𝑥 2) < 𝑡 𝑓 (𝑥 1) + (1 - 𝑡) 𝑓 (𝑥 2)

for all $𝑡 \in [0, 1]$ and $𝑥 1, 𝑥 2 \in 𝑋$ .

Properties

A twice-differentiable function is convex iff its second derivative is nonnegative everywhere
Jensen's inequality
Convexity on the positive reals and $𝑓 (0) \leq 0$ implies superadditivity

#state/tidy | #lang/en | #SemBr