Im not really sure of an exact sized but I used to ride a 148 freeride board (which was pretty narrow) and Im a size 11.5 and I didnt ever have a real problem, but that may be due to my binding set up though. I dont think thats what you were looking for though so Nevermind i guess
If you just cruise groomed trails then a 24.8 would probably be the lowest, whereas you wouldnt catch your overhang on groomed runs as easily.
If you ride a lot of powder you're going to want something like a 25.5+ so your overhang doesnt drag in the snow.
If you ride park you'll want something 25.5 as well because you dont want the overhang on your board to apply too much pressure to the outerside of your edges on boxes and rails because it'll cause you to slip out and catch edges pretty easily.
What is the smallest waist width that can support an 11 boot?
A size 11 foot is 29 cm. The important thing is to find a board that will have reasonable overhang at the inserts (the width at the waist is not a good measurement to use for relating to foot size) at the angles that you ride.
What angles and stance width do you ride? What type of board (park, freeride, etc?).
15 degrees takes off roughly 1 cm (depending a bit on your foot shape). So your 29 cm foot goes down to roughly 28 cm. If you accept that 1 cm of barefoot overhang for each the toe and heel is about the max you would want, then that would bring you down to 26 cm as a minimum. That is actual stance position, not waist. You will need to have the dealer that you are communicating with check the board width at 12 inches off of the waist (to take into account your 24 inch stance) for the models that you are considering.