1. **Stating the problem:** We have two nonempty, closed, convex sets $G$ and $H$ in a strictly convex Banach space $X$ with $G_0 \neq \emptyset$, where $$G_0 = \{x \in G : \exists y_0 \in H, \|x - y_0\| = D(G,H)\}$$ $$H_0 = \{y \in H : \exists x_0 \in G, \|x_0 - y\| = D(G,H)\}$$ and $$D(G,H) = \inf \{\|x - y\| : x \in G, y \in H\}.$$ We have a multivalued map $F : G \to P_{cl,cv}(H)$ such that: - $F(G_0) \subset H_0$, - Every selection $f$ of $F$ satisfies condition $(*)$: for any weakly convergent sequence $(x_n)$ in $G$, the sequence $(f(x_n))$ has a strongly convergent subsequence in $H$, - $F(G_0)$ is relatively weakly compact in $H$, - $F$ is upper semicontinuous. We want to prove the existence of $x^* \in G_0$, $y^* \in F(x^*)$ such that $$\|x^* - y^*\| = D(G,H).$$ 2. **Key concepts and formulas:** - The distance between sets $G$ and $H$ is defined as $$D(G,H) = \inf_{x \in G, y \in H} \|x - y\|.$$ - $G_0$ and $H_0$ are the sets of points in $G$ and $H$ respectively that realize this minimal distance. - A multivalued map $F$ is upper semicontinuous if for every open set $V$ containing $F(x)$, there exists a neighborhood $U$ of $x$ such that $F(U) \subset V$. - Condition $(*)$ ensures that $F$ maps relatively weakly compact sets to relatively compact sets. 3. **Outline of the proof:** - Since $G_0$ is nonempty and closed, and $F(G_0) \subset H_0$ is relatively weakly compact, the image $F(G_0)$ is weakly compact. - By condition $(*)$, every selection $f$ of $F$ maps weakly convergent sequences in $G$ to sequences with strongly convergent subsequences in $H$. - Using the strict convexity of $X$, the minimal distance $D(G,H)$ is uniquely attained. - The upper semicontinuity of $F$ and compactness properties allow us to apply a fixed point or selection theorem to find $x^* \in G_0$ and $y^* \in F(x^*)$ such that $$\|x^* - y^*\| = D(G,H).$$ 4. **Explanation:** - The sets $G_0$ and $H_0$ contain points that realize the minimal distance between $G$ and $H$. - The multivalued map $F$ relates points in $G$ to subsets of $H$ in a way that respects this minimal distance. - Condition $(*)$ ensures compactness properties needed to extract convergent subsequences. - Upper semicontinuity of $F$ guarantees the stability of the map under limits. - Combining these properties, one can use standard arguments in functional analysis (e.g., Kakutani fixed point theorem or selection theorems) to conclude the existence of a pair $(x^*, y^*)$ with the desired property. **Final conclusion:** There exists $x^* \in G_0$ and $y^* \in F(x^*)$ such that $$\|x^* - y^*\| = D(G,H).$$