For every convex body K⊂R^n and for the admissible higher-order indices, the higher-order mixed-volume inequalities proposed by Schneider should hold. These conjectures generalize the classical Godbersen inequality for mixed volumes of K and -K, namely V(K[n-i],(-K)[i])≥ operatornamevol(K) for 0≤ i≤ n, together with the corresponding conjectured characterization of equality. The supplied source does not state the definitions and exact formulas of the two higher-order conjectures, so they cannot be specified more precisely here.

Higher-order Godbersen conjectures

For every convex body $K\subset\mathbb{R}^n$ and for the admissible higher-order indices, the higher-order mixed-volume inequalities proposed by Schneider should hold. These conjectures generalize the classical Godbersen inequality for mixed volumes of $K$ and $-K$ , namely $V(K[n-i],(-K)[i])\geq \operatorname{vol}(K)$ for $0\leq i\leq n$ , together with the corresponding conjectured characterization of equality. The supplied source does not state the definitions and exact formulas of the two higher-order conjectures, so they cannot be specified more precisely here.

References

Primary source

arXiv

Additional references

Around higher-order Godbersen conjectures — arXiv

convex-geometry geometric-inequalities

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed progress

Recent papers claim substantial partial results, including the inequality portion of the higher-order conjectures, but the general conjectures remain open.

The higher-order conjectures generalize Godbersen’s inequality to higher-order difference bodies and mixed volumes. They were proposed in a February 2025 paper; the general inequality and its equality cases are not both settled.

Known results

The stronger higher-order conjecture is proved for anti-blocking convex bodies (February 2025).
Weighted-average inequalities and a separate conjecture for $n \leq 5$ were proved for the original problem (December 2024).
A July 2026 paper claimed a proof of the original Godbersen inequality and stated that its method extends to the higher-order inequality, but deferred the higher-order equality classification.

September 2026 partial results

A September 8, 2026 report describes claimed low-dimensional cases, weighted inequalities, and higher-order join bounds. These constrain possible general proofs but do not settle the higher-order conjectures; the July extension and the new claims remain unverified.

Current status (as of September 2026): Partial results and an unverified claim covering the inequality portion are available, while the general conjectures and higher-order equality characterization remain open.

Sources

No solutions have been posted yet.