OpenAI research collection · 3 manuscripts · Source checked 2026-10-19
A convex shape has a partner called its polar, defined through geometric inequalities. This family claims to identify which shapes minimise the product of their volumes, in every dimension, and describes related inequalities and symplectic geometry results.
The family connects several areas of geometry. Its companion papers prove different statements, so the exact assumptions and formal scope matter.
Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product K× K^\circ in dimension 2n has Gromov width 4.
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝ^(n) has volume product at least 4^n/n!, with equality exactly for invertible linear images of Hanner polytopes.
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We resolve the Mahler conjecture for general convex bodies positively. For every convex body K\subset\mathbb R^n, n ≥ 1, with Santaló point s(K), |K||(K-s(K))^\circ|≥ (n+1)^{n+1}/(n!)^2, with equality exactly for simplices.
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For every integer n ≥ 2 and every origin-symmetric convex body K\subset\mathbb R^n, we prove that the Gromov width of \mathop{int}\nolimits K×\mathop{int}\nolimits K^\circ is 4. We construct smooth symplectic embeddings of standard balls of every capacity 0< c< 4 into this polar product. Volume preservation then resolves the…
For every integer n ≥ 2 and every origin-symmetric convex body K\subset\mathbb R^n, we prove that the Gromov width of \mathop{int}\nolimits K×\mathop{int}\nolimits K^\circ is 4. We construct smooth symplectic embeddings of standard balls of every capacity 0< c< 4 into this polar product. Volume preservation then resolves the symmetric Mahler conjecture positively in every dimension.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
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