Given n≥ 2, a coefficient vector (alpha_1,…,alpha_n)∈R_≥ 0^n setminus0 that is log-concave, meaning alpha_j^2≥ alpha_j-1alpha_j+1 for 2≤ j≤ n-1, and has no internal zeros, and two closed C^2_+ hypersurfaces M,NhookrightarrowR^n+1, does the equality ∑_j=1^n alpha_jE_j(tau_M)=∑_j=1^n alpha_jE_j(tau_N) imply that M and N differ by a translation? The September 2026 preprint claims that the answer is affirmative in this coefficient regime.

Firey's uniqueness question for curvature equations

Given $n\geq 2$ , a coefficient vector $(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}_{\geq 0}^n\setminus\{0\}$ that is log-concave, meaning $\alpha_j^2\geq \alpha_{j-1}\alpha_{j+1}$ for $2\leq j\leq n-1$ , and has no internal zeros, and two closed $C^2_+$ hypersurfaces $\mathcal{M},\mathcal{N}\hookrightarrow\mathbb{R}^{n+1}$ , does the equality $\sum_{j=1}^n\alpha_jE_j(\tau_{\mathcal{M}})=\sum_{j=1}^n\alpha_jE_j(\tau_{\mathcal{N}})$ imply that $\mathcal{M}$ and $\mathcal{N}$ differ by a translation? The September 2026 preprint claims that the answer is affirmative in this coefficient regime.

References

Primary source

arXiv

Additional references

On a question by Firey concerning uniqueness — arXiv

convex-geometry differential-geometry

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to settle Firey’s uniqueness question in a restricted coefficient regime, but its proof has not been refereed.

Firey’s uniqueness question concerns uniqueness for mixed curvature equations and was posed by Firey and Schneider. A new preprint claims an affirmative answer in the stated $C^2_+$ class for nonzero log-concave coefficient vectors with nonnegative entries and no internal zeros.

Known results

For $n=2$ , uniqueness up to translations follows from a theorem of Aleksandrov.
In the general setting, uniqueness for $n\geq 3$ was reported as open in 2023.
Special isotropic and even-solution cases were previously known.

September 2026 claimed resolution

A preprint listed on September 9, 2026 claims to resolve the question in the stated $C^2_+$ class, but the proof is unrefereed and the result is restricted to the specified coefficient regime.

Current status (as of September 2026): A claimed solution covers the stated $C^2_+$ class under the specified coefficient assumptions, while the proof remains unverified and the broader coefficient setting remains open.

Sources

No solutions have been posted yet.