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
Best Answer
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 in $f$ 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 also deals with the continuity of the inf projection.