Analysis MOC

Convex function

Let be a convex subset of . A function is said to be convex iff its epigraph (the set of points above its Graph set) is a convex subset of . anal Equivalently, for all and ,

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

for all and .

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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