When Is the Set of Non-Completable Partial Probability Tensors Convex?

In 2017 Kahle, Kubjas, Kummer and Rosen asked whether, for every pattern of observed entries, the set of nonnegative partial tensors that cannot be completed to the joint distribution of independent discrete random variables is convex. The answer is no, and a negative answer is already implicit in examples of Kubjas and Rosen. We determine exactly when the answer is yes. For tensors of format d_1 × ⋯ × d_n with n ≥ 2 and all d_j ≥ 2, the set is convex if and only if no observed entry is pinned, that is, if through every observed entry there is a maximal slice containing no other observed entry. In that case the completable region is the sublevel set {G_E ≤ 0} of an explicit concave function G_E. Otherwise convexity fails, and under the standing assumptions of Kahle et al. it fails even inside the simplex {Σ_e x_e ≤ 1}. For matrices the condition says that every component of the bipartite graph of observed entries is a star. Under these standing assumptions, to which the question refers, the answer is yes for all patterns only in the formats 2 × m and m × 2; under the two conditions stated with the question alone, only in the format 2 × 2. For n ≥ 3 the convex cases under the standing assumptions are exactly the ∏_j d_j "corner" patterns, which include the running example of Kahle et al. For corner patterns we give a one-variable completability criterion, extending the known cubic criterion for 2 × 2 × 2 tensors, and we show that the irreducible boundary hypersurface has degree at most n(n − 1). This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-15428-015 (Oberwolfach Report 20/2017, "Rank One Tensor Completion", second closing problem).