Sharp Hausdorff Bounds for Singular Sets of Convex Solutions to the k-Hessian Equation

Let 2 ≤ k ≤ n, let Ω ⊂ R^n be open and convex, and let u be a convex viscosity solution of σ_k(D²u) = 1 in Ω. We prove that the set on which u fails to be locally C² has vanishing (n−1)-dimensional Hausdorff measure. In the intermediate range 3 ≤ k < n, this gives a codimension-one refinement of the known almost-everywhere partial regularity, and the exponent is sharp. More generally, for a convex viscosity subsolution of σ_k(D²u) ≥ λ > 0, we obtain Hausdorff bounds for strata defined by the affine dimension of all supporting contact sets. The proof combines a support-dependent Chou–Wang barrier argument, an estimate for the product of the smallest k semiaxes of a John ellipsoid, and Mooney's convex section-covering theorem. In a logically separate structural part, we characterize the distinguished number of flat directions, n−k+1, by an asymptotic infimum mean-value formula over affine sections, and explain how this mean-value heuristic leads to the supporting-contact geometry used in the proof.