Symplectic balls and the symmetric Mahler inequality
OpenAI's theorem (2026) that for every origin-symmetric convex body K in R^n, n >= 2, every symplectic ball of capacity c < 4 embeds into int K x int K°, and its consequence, the symmetric Mahler inequality vol(K) vol(K°) >= 4^n/n!. Polar bodies, support functions and strip bodies; one- and two-forms, the Pfaffian and volume preservation, flows and Moser's method; plurisubharmonic potentials and dd^c; a ball of capacity below pi k from a holomorphic zero of order k; a planar conformal map from the arctangent series and a uniform estimate for its horizontal slice integrals; a symplectic realization inside int K x S int K° with S close to pi k/4, rescaling, and polar approximation of general bodies.
Seven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Polar bodies and the volume product
- Forms, flows and Moser's method
- Plurisubharmonic potentials
- A ball from a holomorphic zero
- A planar conformal map
- The uniform slice estimate
- Realization in a polar product
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Local tools for bundles and transport (Bundles, curvature and holonomy)
- Linear algebra and multivariable calculus, with the inverse and implicit function theorems, as in the core course Real Analysis II (Lebl, Theorems 8.5.1 and 8.5.6).
- Complex analysis of one variable, as in the core course Complex Analysis (Lebl's guide, cited by number).
- Lebesgue measure, dominated convergence, differentiation under the integral sign, Fubini's theorem and the change of variables formula, as in the core course Measure and Integration (Fremlin, cited by number).
- Local solutions of ordinary differential equations and their smooth dependence on initial data: Theorem 2.1 of the lesson Local tools for bundles and transport of the course Bundles, curvature and holonomy; the inverse function theorem, Theorem 1.2 there.