{"id":"openai-math/Slope-field-perturbations-of-the-two-cylinder-covering","pdf":"https://github.com/openai/math/blob/main/preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026","tags":["discrete-geometry","tetrahedra","covering-problems","convex-geometry"],"score":0,"title":"Slope-field perturbations of the two-cylinder covering","domain":"Convex and metric geometry","family":"100","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.MG","summary":"The two-cylinder covering of a regular tetrahedron can be perturbed to give finite covers with total perpendicular base area strictly below half its minimum projection area. These covers give negative answers to both the half-area question and the directionwise normalized half-bound conjecture. By affine invariance, the directionwise conclusion holds for every nondegenerate tetrahedron.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-27","reactions":{},"formalized":false,"family_title":"Cylinder coverings below the half-area bound"}