a little bit of geometry from USSR with love

In the interior of a 12-gonal convex (a polygon with 12 vertices in 2D ) , there exists two points with distance 10cm. Let $O_1,O_2$ , the two interior points and $S(O_1),S(O_2)$ ,be the sum of the distances of the 12 vertical points respectively.Prove that $S(O_1)-S(O_2) < 1m$