I am interested in the volume of the intersection of two Hamming balls of radius say m/6 in m-dimensional space, the distance between whose centers is about \sqrt{m}. I would ideally like this to be a constant or 1/poly(m) fraction of the volume of the Ball itself and wondering if this is true.  This should be an elementary question and I imagine the answer must be well-known, but I am not able to find this on the interweb. Thanks!
Intersection of Hamming Balls
convex geometry