openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Simplex /MetricNormalizationDefinitions.lean
| import OAI.Geometry.CylinderCovering.Simplex.ProjectionDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| namespace CylinderCovering | |
| def RegularMetricNormalizationStatement : Prop := | |
| ∀ v : Fin 4 → Space, AffineIndependent ℝ v → | |
| ∀ edge : ℝ, 0 < edge → | |
| (∀ i j : Fin 4, i ≠ j → dist (v i) (v j) = edge) → | |
| Metric.diam (convexHull ℝ (range v)) = edge ∧ | |
| volume.real (convexHull ℝ (range v)) = edge ^ 3 / (6 * Real.sqrt 2) ∧ | |
| minimumShadowArea (convexHull ℝ (range v)) / 2 = edge ^ 2 / (4 * Real.sqrt 2) | |
| end CylinderCovering | |
| end | |
| end OAI | |