If a compact set is covered by a finite union of open balls of same radii, can we always get a lesser radius?

Replace each open ball $B_i$ of radius $r$ in the cover by the union of concentric open balls of radii strictly smaller than $r$. You get an infinite cover of $V$. By compactness there is a finite subcover. By construction the radii are smaller than before. Finally we choose the maximal radius (for all of the finitely many balls) which is still smaller than $r$.

Let $X$ denote the set of centers: $X = \{c_1,\ldots,c_m\}$.

The function $\phi(x) = \mathop{\rm dist} (x,X)$ is continuous on $\mathbb R^n$ and attains a maximum value on $V$ because $V$ is compact.

Note that if $x \in V$, then by definition $\phi(x) < r$. Whatever maximum it attains must be less than $r$.

Choose $s$ to lie in between this maximum and $r$.