I have searched a lot the prove of the Danes drop Theorem in Banach Spaces but I dont achievement find it. If anyone could tell me where I can read it or know the prove I will be grateful. The statement is the following:
Let $(X, || \cdot ||)$ be a Banach space. Let $A \subset X $ be a nonempty closed and $B \subset X$ a nonempty closed bounded and convex set such that \begin{equation} d(A, B) = \inf \{ || a - b ||: a \in A, b \in B \} > 0 \end{equation} Then, for each $x_0 \in A$ there exists $x^* \in X$ such that: \begin{equation} x^* \in A \cap D(x_0, B) \hspace{.2cm}\text{and}\hspace{.2cm} \{ x^* \} = A \cap D(x^*, B) \end{equation} Note that the "drop" generated by $x \in X$ and $B \subset X$ is defined to be the set \begin{equation} D(x, B) := \{ tx + (1 - t)b: \hspace{.2cm}b \in B, t \in [0, 1] \} \end{equation} Update: The issue now, because of the paper https://dml.cz/bitstream/handle/10338.dmlcz/106385/CommentatMathUnivCarol_026-1985-3_2.pdf is how the generalized K.M. Lemma (Lemma GKZ in paper) implies the Generalized Drop Theorem (the Danes' drop theorem here, Generalized drop theorem in the paper).
A possibility to find the proof would be a pdf-request via scinapse: https://scinapse.io/papers/73432139
A hint on how to prove it is in this paper by Danes: https://dml.cz/bitstream/handle/10338.dmlcz/106385/CommentatMathUnivCarol_026-1985-3_2.pdf
Another option would be to extract a proof from the generalization of Danes drop theorem to locally convex spaces which can be found here: https://www.researchgate.net/publication/254488994_Danes'_Drop_Theorem_in_locally_convex_spaces which, at first glance, seems quite doable without too much time investment.