It can be shown without much difficulty that any Euclidean norms satisfies the following condition :$$(P) \quad B \subset X \subset B' \Rightarrow X \, \text{is convex}$$ where $B=\{x \in E / \|x\| <1\}$ and $\overline B=\{x \in E / \|x\| \leq 1\}$ are respectively the unit open and closed ball of the Eulidean space $(E,\|.\|)$.
I asked myself if $(P)$ is a characterization of Euclidean norms but, looking I found a counter-example.
This is on $\Bbb R^2$: $\displaystyle N(x,y)=\sup_{t \in \Bbb R} \frac{|x+ty|}{1+t+t^2}.$
My questions:
1) is there any studies about the norms satisfying the condition $(P)$ ?
2) I want others counter-examples with norms whose expression is less complicated than the above counter-example.
As Daniel Fischer noted, condition (P) is equivalent to the norm being strictly convex.
Indeed, if the norm is not strictly convex, then the unit sphere contains a line segment, and removing the midpoint of this line segment from the closed unit ball creates a nonconvex set.
Conversely, if the norm is strictly convex, then any $X$ as you described is convex, since all interior points of any line segment with endpoints in $X$ belong to $B$ and therefore to $X$.