Since jet fuel is cheaper, why do pistons still use av-gas?
First off, there are some light aircraft powered by diesel like the Piper Archer DX and the Diamond DA42. It should be noted both come in a 100LL powered version as well. As it stands now the FAA is looking for a replacement for 100LL but has not yet decided/found one. There are also some airplanes out there that can run on mogas. But anyway, more to the point, you cant simply put diesel into a regular engine as the ignition system and compression requirements are different. Thus to convert a plane to diesel you need a different engine all together. With that in mind most certifying bodies dont just let you swap airplane engines to any engine you see fit. The engine would need to be certified for the air frame which is a costly proposition. Thus it is sensible and economical to keep the current engines running. There are some efforts like those made by piper and diamond to slowly make diesel aircraft available. 

As to why 100LL is used, this stems mostly from history. The vast majority of the general aviation fleet both physically and simply in design, stems from the dawn of the jet age. When the lines for the 172 were first drawn there was plenty of 100LL to go around, it was for lack of a better term, the logical choice.