Convex Function - Convex Function Calculus

Convex Function Calculus

• If and are convex functions, then so are and
• If and are convex functions and is non-decreasing, then is convex. As an example, if is convex, then so is, because is convex and monotonically increasing.
• If is concave and is convex and non-increasing, then is convex.
• Convexity is invariant under affine maps: that is, if is convex with, then so is, where
• If is convex in then is convex in provided for some
• If is convex, then its perspective (whose domain is ) is convex.
• The additive inverse of a convex function is a concave function.