Convex antiproximal sets in spaces $c_0$ and $c$

In the note we prove that in a Banach space c there exists a closed bounded symmetric convex division ring $V_1$ such that for any $x\in c\setminus V_1$, $P_{V_1}(x)=\emptyset$ where $P_{V_1}$ is the metric projection onto $V_1$.