Infimal convolution ($E\square F$)

The infimal convolution is $(E\square F)(u)={\inf }_v\{E(v)+F(u-v)\}$. It optimizes a decomposition of $u$ between two costs. It is convex when both costs are convex functions, and its convex conjugate is $E^{\ast }+F^{\ast }$ whenever the extended-real operations are well defined.