I have the 17-40 F4L and since I move the camera while doing video there is sometimes bad distortion on faces... what is the widest I can go <$1000 that has barely any distortion?
FF or crop? The 10-22 on crop has very low distortion.
You might be talking about something different - people "looking fat" near the borders. This is much more noticeable than the "actual" distortion, which is the deviation from the rectangular projection (I am no sure that this is the accepted term) expected from a supposedly distortion free lens.
It's on FF, so most crop options are not viable. 5D3. For example, I had some footage where the face was literally on the side and her face was stretched apart until the camera moved into the frame more.