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I'm reading a book, Set-Valued Analysis, written by Aubin.

I'd like to estimate the Lipschitz constant of the continuous selection of upper hemicontinuous multivalued function by Theorem 9.2.1 of the book. It says that if $F:X \rightarrow Y$ is an upper hemicontinuous multivalued function from a compact metric space $X$ to a Banach space $Y$ and the valued of $F$ are nonempty and convex, then for every $\epsilon > 0$, there exists a locally Lipschitz single valued map $f_{\epsilon}: X \rightarrow Y$ such that $\text{Graph}(f_{\epsilon}) \subset B(\text{Graph}(F), \epsilon)$.

Now I consider the case when $Y$ is also compact, which implies $f_{\epsilon}$ is Lipschitz continuous on $X$. I wonder if we can estimate the Lipschitz constant of such selection ($f_{\epsilon}$) w.r.t. $\epsilon$.

In the proof of the theorem in the paper, the authors construct the selection $f_{\epsilon}$ by introducing locally Lipschitz functions denoted by $a_i:X \rightarrow [0, 1]$. Unfortunately, the authors left a reference for the construction, theorem 1.1.2 in this book, but I cannot found it in the cited book.

Can anybody help me 1) how to construct such $a_i$'s and 2) a way to estimate the Lipschitz constant of $f_{\epsilon}$ if it's possible?

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1 Answer 1

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Actually, my purpose is to show that given a sequence of u.h.c. multivalued functions $F_j:X \rightarrow Y$ we can choose an equi-Lipschitz sequence of functions $f_{\epsilon, j}$ such that $f_{\epsilon, j}$ is an $\epsilon$-selection of $F_j$. Here is my idea if we can actually choose such $a_i$'s described in the proof of the theorem.

As in the proof of theorem 9.2.1. in Set-Valued Analysis (if it's correct), we can choose the finite open cover $ \{ \text{int}(B(x_i, \delta_{x_i}/4) \}_{i \in I}$ of $X$ and $a_i$'s described in the proof, independent of $j$. The selection is given as $f_{\epsilon, j} = \sum_{i \in I} a_i(x) y_i$ with an associated point $y_j \in F_j(B(x_i, \delta_i/4))$. Each $a_i$ is locally Lipschitz in a compact space $X$, then $a_i$ is Lipschitz continuous on $X$ (namely, with Lipschitz constant $L_i$; I don't understand why the authors did not conclude the theorem with Lipschitzness of $f_{\epsilon}$ because local Lipschitzness on a compact space $X$ implies Lipschitzness on the same space. Probably it's due to the generalised version of the theorem as commented above the theorem in the book). Then, for any $x, x' \in X$, $|f_{\epsilon, j}(x) - f_{\epsilon, j}(x)| \leq \max_{x \in X}\lVert x \rVert \cdot \sum_{i \in I}|a_i(x) - a_i(x')| \leq \tilde{L} \lVert x - x'\rVert$ where $\tilde{L} = \max_{x \in X} \lVert x \rVert \sum_{i \in I} L_i$. This implies that $\{ f_{\epsilon, j} \}_{j \in \mathbb{N}}$ is equi-Lipschitz.

My curiosity is quite resolved but I would be appreciated if someone let me know 1) we can actually choose such $a_i$'s and 2) explicit estimation of the Lipschitz constant of $f_{\epsilon}$.

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