Gromov width of symmetric convex polar products
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For an origin-symmetric convex body K⊂Rⁿ with nonempty interior, let K° be its polar. Is the Gromov width of int(K)×int(K°) equal to 4 for every n≥2?
Why it matters and what counts as progress
Why it matters: Equality would turn the known capacity upper bound into an exact symplectic ball-embedding result for every symmetric polar product.
What counts as progress: Symplectic embeddings of every ball with capacity below 4 for all such K, or a polar product whose width is strictly smaller.
How the problem developed
- 2014 · Partial result [1]
Shiri Artstein-Avidan, Roman Karasev, and Yaron Ostrover Showed the Hofer–Zehnder capacity of a symmetric convex body times its polar is 4, giving a related capacity bound but not determining the Gromov width.
- 22 Sept 2026 · Reported claim [2]
OpenAI The preprint claims that every origin-symmetric convex body K in dimension n≥2 has Gromov width 4 for the interior of K×K°, with no embedding at capacity 4.
AI contribution
Symplectic Balls in Symmetric Polar Products
For every origin-symmetric compact convex body K with nonempty interior in Rⁿ, n≥2, the Gromov width of int(K)×int(K°) is 4. Every symplectic ball of capacity c<4 embeds.
Sources
- Artstein-Avidan, Karasev, and Ostrover, From symplectic measurements to the Mahler conjecture
- OpenAI, Symplectic balls in polar products
- Symplectic Balls in Symmetric Polar Products
- Symplectic balls in symmetric polar products
- OpenAI’s formal theorem scope
- Lean proof of polar-product width
- OpenAI mathematics release provenance
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