2026/07/31 by Xiyu Hu
Mathematics · #math.AP #msc:35J60 #msc:35B65 #msc:35D40 #msc:28A78 #msc:52A20
34 pages, 2 figures. Comments, questions, and corrections are welcome. A bilingual expository note with interactive figures is available at https://hxypqr.github.io/post.html?id=113
arxiv created 2026/07/31 · arxiv updated 2026/08/03
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<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 σk(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. As a direct analytical consequence, the full distributional Hessian is absolutely continuous and u∈ W2,1loc(Ω), yielding a k-Hessian counterpart of the W2,1 regularity known for singular Monge--Ampère solutions. 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.