Relaxing the convexity assumption on the ellipticity set

Determine whether the convexity assumption on the ellipticity set can be relaxed in the characterization of Hessian operators satisfying the smooth integral comparison principle.

Background

The main characterization theorem assumes that the ellipticity set is open and convex: under this hypothesis, a Hessian operator satisfying the integral comparison principle must be a linear combination of elementary Hessian operators. The authors note that convexity is used crucially in the proof of the sufficiency direction, while their argument for necessity does not require it.

The remark identifies the unresolved issue of whether convexity is genuinely necessary for the result or merely a technical assumption. Resolving this would extend the integral-comparison-principle classification to broader classes of ellipticity sets.

References

In our argument we shall make crucial use of convexity but we do not know whether it can be relaxed.

— Hessian operators, null Lagrangians and the integral comparison principle  (2610.06313 - Dinew et al., 5 Oct 2026) in Remark in Section 2, immediately following Definition of ellipticity sets

However, to our knowledge, there is no known characterization of the convexity of ellipticity sets for general operator of the form affcomb.

— Hessian operators, null Lagrangians and the integral comparison principle  (2610.06313 - Dinew et al., 5 Oct 2026) in Section 1, paragraph following Proposition 1.3