Soprunov–Zvavitch conjecture

The conjecture asks whether a certain mixed-volume inequality characterizes simplices among convex bodies. Earlier work established important partial cases, while a new preprint claims the full characterization.

Known results

• Soprunov–Zvavitch (2015–2016): proved indecomposability, settling simple polytopes and the two-dimensional case.
• Saroglou–Soprunov–Zvavitch (2015–2016): proved the characterization for convex $n$-polytopes and derived boundary obstructions for general convex bodies.
• Later work confirmed the three-dimensional case and developed further necessary conditions, while describing dimensions $n \ge 4$ as open.

September 17, 2026 all-dimensional proof claim

Dylan Langharst and Shouda Wang’s arXiv preprint claims two proofs of the conjectured characterization in every dimension, together with related chord and inradius characterizations. This would close the remaining general case, but the claim is currently unverified.

Current status (as of September 2026): Earlier partial cases are established, and an arXiv preprint claims an all-dimensional solution, but the complete conjecture remains unverified.