If your 2D convex polygons are defined by an array of 2D Vectors then you are looking for line-line intersection or edge-edge intersection, whichever you prefer.
This stackoverflow page has an advanced and precise algorith: http://stackoverflow.com/questions/2255842/detecting-coincident-subset-of-two-coincident-line-segments/2255848#2255848https://stackoverflow.com/questions/2255842/detecting-coincident-subset-of-two-coincident-line-segments/2255848#2255848
