Let $N$ be a hypersurface in $\mathbb R^n$, assume it is compact. Then the maximum point of $d(O, x)$ when restrict to $N$ has positive sectional curvature lower bound by the one of the correspond sphere tangent to it, right?

The best lower bound would be the smallest sphere contains $N$, right?

I am asking this since it seems someone estimates the Ricci curvature of such point, a natural question is why not estimate the sectional curvature if what I said is correct?