Convex conjugate

Convexity

For two functions and and a number the convexity relation

holds. The operation is a convex mapping itself.

The infimal convolution (or epi-sum) of two functions f and g is defined as

Let f₁, …, f_m be proper, convex and lsc functions on R. Then the infimal convolution is convex and lsc (but not necessarily proper), and satisfies