Editing Convex conjugate (section)

=== Infimal convolution ===
The '''infimal convolution''' (or epi-sum) of two functions <math>f</math> and <math>g</math> is defined as

:<math>\left( f \operatorname{\Box} g \right)(x) = \inf \left\{ f(x-y) + g(y) \mid y \in \mathbb{R}^n \right\}.</math>

Let <math>f_1, \ldots, f_{m}</math> be [[Proper convex function|proper]], convex and [[Semi-continuity|lower semicontinuous]] functions on <math>\mathbb{R}^{n}.</math> Then the infimal convolution is convex and lower semicontinuous (but not necessarily proper),<ref>{{cite book |last=Phelps |first=Robert |authorlink=Robert R. Phelps |title=Convex Functions, Monotone Operators and Differentiability|url=https://archive.org/details/convexfunctionsm00phel |url-access=limited | edition=2 |year=1993|publisher=Springer |isbn= 0-387-56715-1|page= [https://archive.org/details/convexfunctionsm00phel/page/n50 42]}}</ref> and satisfies

:<math>\left( f_1 \operatorname{\Box} \cdots \operatorname{\Box} f_m \right)^{*} = f_1^{*} + \cdots + f_m^{*}.</math>

The infimal convolution of two functions has a geometric interpretation:  The (strict) [[epigraph (mathematics)|epigraph]] of the infimal convolution of two functions is the [[Minkowski sum]] of the (strict) epigraphs of those functions.<ref>{{cite journal |doi=10.1137/070687542 |title=The Proximal Average: Basic Theory |year=2008 |last1=Bauschke |first1=Heinz H. |last2=Goebel |first2=Rafal |last3=Lucet |first3=Yves |last4=Wang |first4=Xianfu |journal=SIAM Journal on Optimization |volume=19 |issue=2 |pages=766|citeseerx=10.1.1.546.4270 }}</ref>