Quicksort still requires O(lg n) memory because of its recursive nature. Since (lg n) is generally small, it's not that big of a deal. But with bad data, it's possible for Quicksort to degenerate to O(n) extra memory, which could be an issue (Especially for machines with little stack space).
Heapsort (And thus smoothsort) are not recursive, so they require a constant amount of memory for all inputs.
What if you always recurse into the smaller half, and use tail-recursion/looping to process the rest? Doesn't that keep the space usage bounded to O(log(n))?
