Let $(V, \lVert \cdot \rVert)$ be a normed space. For Let $O \colon 2^V \to 2^V$ be defined by the formula: \begin{equation} \forall A \in 2^V \quad O(A) = \bigcap_{x \in V} \overline{B}(x, r_A(x)), \end{equation} where $\overline{B}(x, r) = \{ y \in V \colon \lVert y - x \rVert \le r\}$ and $r_A(x) = \sup \{ \lVert a - x \rVert \colon a \in A\}$. Clearly, for every $A \subseteq V$ set $O(A)$ is convex. In the case when $A$ is (metrically) unbounded, $O(A) = V$, because one call allow for $r_A(x)$ to equal $\infty$. In the case when $A = \{x\}$, then $O(A) = \{x\}$.

Question:

I am interested in whether there is some characterization of what types of sets must be in the range of $O$.

For example, if $V = \mathbb{R}^2$, I suspect that regardless of the used norm, the range of $O$ will contain all line segments which are parallel to at least two noncollinear directions. Here, the reason being that:

• if a given direction is parallel to the supporting line of the unit ball at some point and the supporting line is unique, then that directionhas the mentioned property,
• if at a given point of the unit ball we have some family of supporting lines, then the directions from the boundary of this family, here, lines with the highest and lowest leading coefficient in the slope-intercept form, have this property