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

Let 2 ≤ k ≤ n, let Ω ⊂ Rⁿ be open and convex, and let u be a convex viscosity solution of σₖ (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 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.