For a seminar I am working on a Moreau-Yosida regularization in Banach spaces.


The regularization is defined by


$$f\_\lambda(x) := \inf \left \{ \frac{\|x-y\|^2}{2\lambda} +f(y) : y \in X \right \}, ~ x \in X$$


where $X$ is a reflexive and strictly convex Banach space, $f: X \rightarrow \mathbb{R} \cup \{+\infty\}$ lower-semicontinuous, proper and convex, $\lambda > 0$.


In **Convexity and Optimization in Banach Spaces**, V. Barbu and T. Precupanu, Springer, 2012, the authors prove that the infimum defining $f\_\lambda(x)$ is attained for all $x \in X$. 


From that they deduce that $f\_\lambda$ is convex and lower-semicontinious, but, unfortunately, they don't elaborate the argument. However, I do not see how it follows from what is given.


As far as I am aware, most literature deals with the case where $X$ is Hilbert, I could not finde the result stated above for the general case in another publication.



> 
> **Question:** Maybe someone can recommed me a paper where this result is proven? Or is there an obvious argument which I am missing at the moment?
> 
> 
> 


Thank you in advance



Moreau-Yosida regularization in Banach spaces
I deleted a previous wrong and misleading answer.


The Moreau-Yoshida envelope is a special case of th infimal convolution of two convex functions $f$ and $g$ which is defined as
$$
f\Box g(x) = \inf\_{y\in Y} f(y) + g(x-y).
$$
In other words: The infimal convolution of $f$ and $g$ is the largest (extended) real valued functional whose epigraph contains the (Minkowski) sum of the epigraphs of $f$ and $g$. Consequently, it is convex and lsc if $f$ and $g$ are.


A more verbatim argument for the convexity: In general if $F:X\times X\to ]-\infty,\infty]$ is convex, then $f(x) = \inf\_y F(x,y)$ is convex: Take $x\_1$ and $x\_2$ such that $f(x\_i)$ is finite and $\xi\_1> f(x\_1)$, $\xi\_2> f(x\_2)$. Then there exist $y\_1$, $y\_2$ such that $F(x\_i,y\_i)<\xi\_i$. By convexity of $F$ it holds for $0<\lambda<1$ that
$$\begin{array}{rl}
f(\lambda x\_1 + (1-\lambda)x\_2) & \leq F(\lambda x\_1 + (1-\lambda)x\_2, \lambda y\_1 + (1-\lambda)y\_2)\\
& \leq \lambda F(x\_1,y\_1) + (1-\lambda)F(x\_2,y\_2)\\
& \leq \lambda \xi\_1 + (1-\lambda)\xi\_2.
\end{array}
$$
Letting $\xi\_i\to f(x\_i)$ shows convexity of $f$. Now apply this to $F(x,y) = f(y) + g(x-y)$.


By the way: One calls $f$ the *inf-projection* of $F$. The paper "[Lipschitz Continuity of inf-Projections" by Roger J.-B. Wets](http://www.math.ucdavis.edu/%7Erjbw/mypage/Variational_Analysis_files/Wets03_infpr.pdf) also deals with the continuity of the inf projection.


