Common Support Lines of Three Disjoint Convex Sets Have Zero Volume

Let A, B, C be pairwise disjoint convex subsets of R^3. We prove that the union of the lines supporting all three sets has Lebesgue outer measure zero. No smoothness, strict convexity, boundedness, closedness, or dimension assumption is required. A bi-Lipschitz chart of lines with two distinct simple support contacts is obtained from local convex graphs and a contraction argument. At density points of the parameter set giving a third contact, the three support-normal constraints force parallel normals. The exceptional lines are covered by Ivanov's theorem on extended boundary segments. Countable enlargement and localization give the result for arbitrary convex sets. This establishes the measure-zero assertion in Goaoc's 2008 problem, corresponding to OWR-2090-028 in ulamai/UnsolvedMath. The stronger countable 2-manifold-cover assertion is not addressed. This is an AI-assisted, self-audited, unrefereed preprint; no independent review, formal certification, or exhaustive priority determination is claimed.