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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.

For $z\in {\mathbb{R}}^m$ and $\beta >0$, the scaled log-sum-exp function isIt is a smooth convex function satisfying

A monotone operator is maximal when its graph has no proper enlargement that is still monotone. In Euclidean or Hilbert space, its resolvent of a monotone operator is single-valued and everywhere defined for every positive step. Subdifferentials of proper lower semicontinuous convex functions are fundamental examples.

This convex function penalizes positive densities using negative entropy, with $0\log 0=0$ and infinite value for negative inputs. Its scalar derivative is $\log x$ on $x>0$. On a bounded domain, restricting to uniformly positive bounded functions gives a simple setting for a Frechet derivative. Its Bregman distance is the generalized Kullback–Leibler divergence. With fixed total mass, minimizing this negative-entropy expression maximizes the corresponding entropy.

An operator $F$ on an inner-product space is monotone whenfor every $v,w$. The gradient of every differentiable convex function is monotone.