Short··0:20·Convex and metric geometry
Simplex Conjecture for Isotropic Constants: OpenAI Claims a Proof
Watch on YouTubeAbout the problem →
About this video
Squash any convex solid, keeping its volume, until it spins evenly in every direction: that is isotropic position, and how spread out the solid then is gives its isotropic constant. The strong isotropic constant conjecture says the simplex has the largest one in every dimension. OpenAI claims a proof that simplices are the unique maximizers, together with a sharp entropy bound for log-concave densities. The animation computes the exact isotropic constants of 22 solids in three dimensions (cube, ball, cylinder, cone, octahedron and random polytopes); none beats the tetrahedron's 0.3225. This is OpenAI's preprint, not yet peer reviewed.
Paper (OpenAI, catalogue result 101):
- A sharp entropy bound and the simplex inequality for isotropic constants: github.com/openai/math/blob/main/preprints/A-sharp-entropy-bo…
The animation illustrates the result; it is not a proof.
Gradient Damage: mathematics you can see, equations you can hear.
youtube.com/@gradientdamage
Every result explained: gradientdamage.com/p/101
About the problem
Convex and metric geometry·OpenAI Math·Result 101
The sharp simplex conjecture for isotropic constants
Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.
Topics
isotropic constantsimplex conjecturestrong isotropic constant conjectureslicing problemBourgain slicingconvex bodiesisotropic positionlog-concaveentropy boundconvex geometryasymptotic geometric analysissimplextetrahedronOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animation
