Analysis MOC

Convex function

Let 𝑋 be a convex subset of ℝ𝑛. A function 𝑓 :𝑋 →ℝ is said to be convex iff its epigraph (the set of points above its Graph set) is a convex subset of ℝ𝑛+1. anal Equivalently, for all 𝑡 ∈[0,1] and 𝑥1,𝑥2 ∈𝑋,

𝑓(𝑡𝑥1+(1−𝑡)𝑥2)≤𝑡𝑓(𝑥1)+(1−𝑡)𝑓(𝑥2)

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(𝑡𝑥1+(1−𝑡)𝑥2)<𝑡𝑓(𝑥1)+(1−𝑡)𝑓(𝑥2)

for all 𝑡 ∈[0,1] and 𝑥1,𝑥2 ∈𝑋.

Properties

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

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