There are two main schools of thought when it comes to the position of the thumb: the thumb towards the left of the neck, or right under it.
The following is not in a lesson format, but rather a discussion as to what works for me, and which could be useful in whole or in part, to others. I would like to accredit Tony Devroye of the Avalon Quartet for his help in this issue.
This is the way I've been holding the instrument for years and years. Recently, it has been seen that this can cause two complications:
1. Clamping with the hand can cut down circulation, which is why playing Kreutzer No. 9 three times through can cause eventual pain, especially in higher positions, and when the 4th finger is used.
2. The vibrato can be limited.
3. The thumb does extra work when shifting.
So, the thing to do is to have some clearance on the right side, and move the thumb under the instrument. A side effect of that is that the wrist can jut out (highlighted area), making all the fingers unstable on the string, resulting in funky intonation. You may also notice the huge space created by the rounding of the hand.
Nevertheless, the benefit is that it covers all the three goals mentioned earlier. It's just too wide. So the next thing to do is to push the wrist inwards, making the arm straight.
And that's the result - the fingers might not want to be that high from the string, but that is just to make the view clearer.
1. The space helps in easing physical strain
2. The thumb helps eventually, but a lot of scales up and down single strings need to be done to get used to shifting. Scales need to be practiced with speed, and with the use of the 4th finger.
That's the easy part. The difficult part is when one tries to do the same thing on the lower strings, especially if you have a thumb that goes entirely upright and cannot bend further back (i.e. not double-jointed). This will prevent the left hand from moving from Stage B to Stage C on the lower strings without angling the thumb on the neck of the instrument in a very awkward way. Keeping it in Stage B is still good for vibrato, but causes complications for shifting.
So: for lower strings at least, it is a reasonable compromise to keep the thumb back in Stage A at least for the lower strings, but avoid clamping the neck of the instrument. When it is possible to move the thumb under the neck, it can have interesting benefits and it is good to have that movement in the vocabulary of the left hand, and be able and comfortable in using it, even if one decides not to.
The final analysis (or as my prof says, the bottom line): whichever way the thumb is positioned, make sure the area of the wrist closest to the instrument is relaxed.