I need the mic to be close to the instrument because I don't record in a soundproof studio.

I'm no guru when it comes to violins and such, but I'll offer this:
Many great recordings have been done in living rooms and such.
Where do you record?

The object of a recording studio's acoustics are not so much soundproofing as it is sound control - but if you mean cutting the sound bleeding into or out of the room, that's an issue that's difficult to solve without $$.

Move the mic away from the violin, just a little bit, angle it a little to catch some room sound, record.
If it still doesn't sound right, move it a little more.

Try putting yourself and the mic in a spot where you can get minimal direct reflections from the surrounding walls, windows, and any large dense pieces of furniture. If you can hang a blanket or comforter across a broad section of wall, or large window, do that.

I like to use a 404 or 4050 about 3 feat from the violin at a 50 degree angle.
the wave length has completed well over one full cycle, so it sounds nice and full. plus you get some of the room noise which adds a bit of well needed softness to the tone.

I'm by no means seasoned at recording a violin, but I once heard someone say (paraphrasing) "Recording the violin and recording the SOUND of a violin are two completely separate, and drastic tasks".

He went on to explain that you should record it 3,4,5,6 etc. feet away.

Ribbon microphones work well on strings.

Best bet is to get the violinist to play while you walk around him or her with one ear covered over by your hand. Go around until you find the best sweet spot listening with the uncovered ear pointing towards the violin (it will be readily apparent where the best sweet spot is) and then put the microphone where your ear was and do some additional tweaking.

Works every time.

I tried to record the violin putting the mic 5 cm beneath the instrument. The high frequencies are less strong and the sound softer. I am wondering which is the precise pattern the frequencies follow spreading outside the violin. Any help?

Try these references...

http://emusician.com/tutorials/Resonance_and_Radiation/

http://www.acoustics.org/press/131st/lay12.html

Here's another option.

Every violin is slightly different. Use your EARS to determine where to place the microphone.