A Note on the Size of the Largest Ball Inside a Convex Polytope

Let $m>1$ be an integer, $B_m$ the set of all unit vectors of $\Bbb R^m$ pointing in the direction of a nonzero integer vector of the cube $[-1, 1]^m$. Denote by $s_m$ the radius of the largest ball contained in the convex hull of $B_m$. We determine the exact value of $s_m$ and obtain the asymptotic equality $s_m\sim\frac{2}{\sqrt{\log m}}$.