Really.
1) “The world” is not convinced of the resurrection. In fact, many parts of the world where it used to be believed no longer do so.
2) How do you account for the many apparently false beliefs that are widely accepted?
The known world at the time in question was certainly converted to Christianity on the belief of the resurrection, so they were convinced. As to the fact that some people no longer accept that fact, so what? The evidence is still beyond doubt. Many refused to accept the evidence in the ancient world also.
"2) How do you account for the many apparently false beliefs that are widely accepted?"
Fallen man, and lack of having been evangelized..